From 01ea45b763a3742768d20cf9533ab70ca6b0c89e Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 30 Aug 2025 22:47:59 +0200
Subject: [PATCH 01/41] renaming polynomial endofunctors

---
 ...unctoriality-dependent-pair-types.lagda.md | 57 +++++++----
 .../transport-along-identifications.lagda.md  |  5 +
 ...equality-dependent-function-types.lagda.md | 10 ++
 src/foundation/function-types.lagda.md        |  4 +-
 src/foundation/path-cosplit-maps.lagda.md     |  2 +-
 src/foundation/slice.lagda.md                 |  2 +-
 src/foundation/type-duality.lagda.md          |  4 +-
 .../algebras-polynomial-endofunctors.lagda.md |  4 +-
 ...lgebra-of-enriched-directed-trees.lagda.md |  2 +-
 ...oalgebras-polynomial-endofunctors.lagda.md |  4 +-
 ...-algebras-polynomial-endofunctors.lagda.md | 16 ++--
 ...oalgebras-polynomial-endofunctors.lagda.md | 16 ++--
 src/trees/polynomial-endofunctors.lagda.md    | 95 ++++++++++++++-----
 src/trees/w-types.lagda.md                    | 20 ++--
 14 files changed, 162 insertions(+), 79 deletions(-)

diff --git a/src/foundation-core/functoriality-dependent-pair-types.lagda.md b/src/foundation-core/functoriality-dependent-pair-types.lagda.md
index f3d941d326..30b3e05d74 100644
--- a/src/foundation-core/functoriality-dependent-pair-types.lagda.md
+++ b/src/foundation-core/functoriality-dependent-pair-types.lagda.md
@@ -55,8 +55,7 @@ Any family of maps induces a map on the total spaces.
 
 ```agda
   tot : Σ A B → Σ A C
-  pr1 (tot t) = pr1 t
-  pr2 (tot t) = f (pr1 t) (pr2 t)
+  tot t = (pr1 t , f (pr1 t) (pr2 t))
 ```
 
 ### Any map `f : A → B` induces a map `Σ A (C ∘ f) → Σ B C`
@@ -67,8 +66,7 @@ module _
   where
 
   map-Σ-map-base : Σ A (λ x → C (f x)) → Σ B C
-  pr1 (map-Σ-map-base s) = f (pr1 s)
-  pr2 (map-Σ-map-base s) = pr2 s
+  map-Σ-map-base s = (f (pr1 s) , pr2 s)
 ```
 
 ### The functorial action of dependent pair types, and its defining homotopy
@@ -79,10 +77,8 @@ module _
   (D : B → UU l4)
   where
 
-  map-Σ :
-    (f : A → B) (g : (x : A) → C x → D (f x)) → Σ A C → Σ B D
-  pr1 (map-Σ f g t) = f (pr1 t)
-  pr2 (map-Σ f g t) = g (pr1 t) (pr2 t)
+  map-Σ : (f : A → B) (g : (x : A) → C x → D (f x)) → Σ A C → Σ B D
+  map-Σ f g t = (f (pr1 t) , g (pr1 t) (pr2 t))
 
   triangle-map-Σ :
     (f : A → B) (g : (x : A) → C x → D (f x)) →
@@ -123,7 +119,7 @@ tot-htpy H (pair x y) = eq-pair-eq-fiber (H x y)
 tot-id :
   {l1 l2 : Level} {A : UU l1} (B : A → UU l2) →
   (tot (λ x (y : B x) → y)) ~ id
-tot-id B (pair x y) = refl
+tot-id B p = refl
 ```
 
 ### The map `tot` preserves composition
@@ -134,7 +130,7 @@ preserves-comp-tot :
   {A : UU l1} {B : A → UU l2} {B' : A → UU l3} {B'' : A → UU l4}
   (f : (x : A) → B x → B' x) (g : (x : A) → B' x → B'' x) →
   tot (λ x → (g x) ∘ (f x)) ~ ((tot g) ∘ (tot f))
-preserves-comp-tot f g (pair x y) = refl
+preserves-comp-tot f g p = refl
 ```
 
 ### The fibers of `tot`
@@ -414,7 +410,8 @@ module _
 equiv-Σ-equiv-base :
   {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (C : B → UU l3) (e : A ≃ B) →
   Σ A (C ∘ map-equiv e) ≃ Σ B C
-pr1 (equiv-Σ-equiv-base C (pair f is-equiv-f)) = map-Σ-map-base f C
+pr1 (equiv-Σ-equiv-base C (pair f is-equiv-f)) =
+  map-Σ-map-base f C
 pr2 (equiv-Σ-equiv-base C (pair f is-equiv-f)) =
   is-equiv-map-Σ-map-base f C is-equiv-f
 ```
@@ -477,13 +474,13 @@ module _
 
   fiber-triangle :
     (x : X) → fiber f x → fiber g x
-  pr1 (fiber-triangle .(f a) (pair a refl)) = h a
-  pr2 (fiber-triangle .(f a) (pair a refl)) = inv (H a)
+  pr1 (fiber-triangle .(f a) (a , refl)) = h a
+  pr2 (fiber-triangle .(f a) (a , refl)) = inv (H a)
 
   square-tot-fiber-triangle :
     ( h ∘ map-equiv-total-fiber f) ~
     ( map-equiv-total-fiber g ∘ tot fiber-triangle)
-  square-tot-fiber-triangle (pair .(f a) (pair a refl)) = refl
+  square-tot-fiber-triangle (.(f a) , a , refl) = refl
 ```
 
 ### In a commuting triangle, the top map is an equivalence if and only if it induces an equivalence on fibers
@@ -560,10 +557,36 @@ module _
   (g : (a : A) → X a → Y a) {a a' : A} {x : X a} {x' : X a'}
   where
 
-  compute-ap-tot :
+  coh-ap-tot :
     pair-eq-Σ ∘ ap (tot g) {a , x} {a' , x'} ~
-    tot (λ p q → inv (preserves-tr g p x) ∙ ap (g a') q) ∘ pair-eq-Σ
-  compute-ap-tot refl = refl
+    tot (λ p q → inv-preserves-tr g p x ∙ ap (g a') q) ∘ pair-eq-Σ
+  coh-ap-tot refl = refl
+
+  compute-ap-tot :
+    (p : a ＝ a') →
+    (q : dependent-identification X p x x') →
+    ap (tot g) {a , x} {a' , x'} (eq-pair-Σ p q) ＝
+    eq-pair-Σ p (inv-preserves-tr g p x ∙ ap (g a') q)
+  compute-ap-tot refl refl = refl
+
+  compute-ap-tot' :
+    (p : (a , x) ＝ (a' , x')) →
+    ap (tot g) {a , x} {a' , x'} p ＝
+    eq-pair-Σ
+      ( ap pr1 p)
+      ( inv-preserves-tr g (ap pr1 p) x ∙
+        ap (g a') (tr-ap pr1 (λ x _ → pr2 x) p a))
+  compute-ap-tot' refl = refl
+
+  inv-compute-ap-tot' :
+    (p : (a , x) ＝ (a' , x')) →
+    eq-pair-Σ
+      ( ap pr1 p)
+      ( inv-preserves-tr g (ap pr1 p) x ∙
+        ap (g a') (tr-ap pr1 (λ x _ → pr2 x) p a)) ＝
+    ap (tot g) {a , x} {a' , x'} p
+  inv-compute-ap-tot' p = inv (compute-ap-tot' p)
+
 ```
 
 ### Computing the action on identifications of the functorial action of Σ
diff --git a/src/foundation-core/transport-along-identifications.lagda.md b/src/foundation-core/transport-along-identifications.lagda.md
index 49b511e7e0..c808e18233 100644
--- a/src/foundation-core/transport-along-identifications.lagda.md
+++ b/src/foundation-core/transport-along-identifications.lagda.md
@@ -63,6 +63,11 @@ module _
     f j (tr A p x) ＝ tr B p (f i x)
   preserves-tr refl x = refl
 
+  inv-preserves-tr :
+    {i j : I} (p : i ＝ j) (x : A i) →
+    tr B p (f i x) ＝ f j (tr A p x)
+  inv-preserves-tr p x = inv (preserves-tr p x)
+
   compute-preserves-tr :
     {i j : I} (p : i ＝ j) (x : A i) →
     preserves-tr p x ＝
diff --git a/src/foundation/equality-dependent-function-types.lagda.md b/src/foundation/equality-dependent-function-types.lagda.md
index 04e62574f7..5a58c07765 100644
--- a/src/foundation/equality-dependent-function-types.lagda.md
+++ b/src/foundation/equality-dependent-function-types.lagda.md
@@ -54,6 +54,16 @@ module _
         ( equiv-explicit-implicit-Π)
         ( λ _ → equiv-explicit-implicit-Π))
       ( is-torsorial-Eq-Π)
+
+
+  is-torsorial-Eq-implicit-Π' : is-torsorial (λ g → (x : A) → C x (g {x}))
+  is-torsorial-Eq-implicit-Π' =
+    is-contr-equiv
+      ( Σ ((x : A) → B x) (λ g → (x : A) → C x (g x)))
+      ( equiv-Σ-equiv-base
+        ( λ g → (x : A) → C x (g x))
+        ( equiv-explicit-implicit-Π))
+      ( is-torsorial-Eq-Π)
 ```
 
 ### Extensionality
diff --git a/src/foundation/function-types.lagda.md b/src/foundation/function-types.lagda.md
index d6fb4ed16f..600fdfb138 100644
--- a/src/foundation/function-types.lagda.md
+++ b/src/foundation/function-types.lagda.md
@@ -91,8 +91,8 @@ module _
 
   compute-dependent-identification-function-type :
     (p : x ＝ y) (f : B x → C x) (g : B y → C y) →
-    ((b : B x) → tr C p (f b) ＝ g (tr B p b)) ≃
-    dependent-identification (λ a → B a → C a) p f g
+    ( (b : B x) → tr C p (f b) ＝ g (tr B p b)) ≃
+    ( dependent-identification (λ a → B a → C a) p f g)
   compute-dependent-identification-function-type refl f g =
     inv-equiv equiv-funext
 
diff --git a/src/foundation/path-cosplit-maps.lagda.md b/src/foundation/path-cosplit-maps.lagda.md
index 854352b42a..1e86bc9b54 100644
--- a/src/foundation/path-cosplit-maps.lagda.md
+++ b/src/foundation/path-cosplit-maps.lagda.md
@@ -507,7 +507,7 @@ is-path-cosplit-tot {k = succ-𝕋 k} {f = f} F x y =
   is-path-cosplit-equiv-arrow
     ( equiv-pair-eq-Σ x y ,
       equiv-pair-eq-Σ (tot f x) (tot f y) ,
-      compute-ap-tot f)
+      coh-ap-tot f)
     ( is-path-cosplit-tot
       { f = λ p q → inv (preserves-tr f p (pr2 x)) ∙ ap (f (pr1 y)) q}
       ( λ where refl → F (pr1 y) (pr2 x) (pr2 y)))
diff --git a/src/foundation/slice.lagda.md b/src/foundation/slice.lagda.md
index 0a41b6dd82..cf5d855706 100644
--- a/src/foundation/slice.lagda.md
+++ b/src/foundation/slice.lagda.md
@@ -52,7 +52,7 @@ definitions for types.
 
 ```agda
 Slice : (l : Level) {l1 : Level} (A : UU l1) → UU (l1 ⊔ lsuc l)
-Slice l = type-polynomial-endofunctor (UU l) (λ X → X)
+Slice l = type-polynomial-endofunctor' (UU l) (λ X → X)
 ```
 
 ### The morphisms in the slice category of types
diff --git a/src/foundation/type-duality.lagda.md b/src/foundation/type-duality.lagda.md
index c2beea330e..d29f7756a8 100644
--- a/src/foundation/type-duality.lagda.md
+++ b/src/foundation/type-duality.lagda.md
@@ -73,7 +73,7 @@ type-polynomial-endofunctor-UU l = Slice l
 map-polynomial-endofunctor-UU :
   (l : Level) {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) →
   type-polynomial-endofunctor-UU l A → type-polynomial-endofunctor-UU l B
-map-polynomial-endofunctor-UU l = map-polynomial-endofunctor (UU l) (λ X → X)
+map-polynomial-endofunctor-UU l = map-polynomial-endofunctor' (UU l) (λ X → X)
 ```
 
 ### Type families
@@ -90,7 +90,7 @@ map-exp-UU l f P = P ∘ f
 
 ## Properties
 
-### If `A` is locally `l`-small, then we can construct an embedding `type-polynomial-endofunctor l A ↪ type-exp-UU A`
+### If `A` is locally `l`-small, then we can construct an embedding `type-polynomial-endofunctor' l A ↪ type-exp-UU A`
 
 ```agda
 map-type-duality :
diff --git a/src/trees/algebras-polynomial-endofunctors.lagda.md b/src/trees/algebras-polynomial-endofunctors.lagda.md
index ca1ac4e432..bd2daee03f 100644
--- a/src/trees/algebras-polynomial-endofunctors.lagda.md
+++ b/src/trees/algebras-polynomial-endofunctors.lagda.md
@@ -29,7 +29,7 @@ algebra-polynomial-endofunctor :
   (l : Level) {l1 l2 : Level} (A : UU l1) (B : A → UU l2) →
   UU (lsuc l ⊔ l1 ⊔ l2)
 algebra-polynomial-endofunctor l A B =
-  Σ (UU l) (λ X → type-polynomial-endofunctor A B X → X)
+  Σ (UU l) (λ X → type-polynomial-endofunctor' A B X → X)
 
 type-algebra-polynomial-endofunctor :
   {l l1 l2 : Level} {A : UU l1} {B : A → UU l2} →
@@ -39,7 +39,7 @@ type-algebra-polynomial-endofunctor X = pr1 X
 structure-algebra-polynomial-endofunctor :
   {l l1 l2 : Level} {A : UU l1} {B : A → UU l2}
   (X : algebra-polynomial-endofunctor l A B) →
-  type-polynomial-endofunctor A B (type-algebra-polynomial-endofunctor X) →
+  type-polynomial-endofunctor' A B (type-algebra-polynomial-endofunctor X) →
   type-algebra-polynomial-endofunctor X
 structure-algebra-polynomial-endofunctor X = pr2 X
 ```
diff --git a/src/trees/coalgebra-of-enriched-directed-trees.lagda.md b/src/trees/coalgebra-of-enriched-directed-trees.lagda.md
index 32cae5c137..36112c9f2a 100644
--- a/src/trees/coalgebra-of-enriched-directed-trees.lagda.md
+++ b/src/trees/coalgebra-of-enriched-directed-trees.lagda.md
@@ -38,7 +38,7 @@ module _
 
   structure-coalgebra-Enriched-Directed-Tree :
     Enriched-Directed-Tree l3 l3 A B →
-    type-polynomial-endofunctor A B (Enriched-Directed-Tree l3 l3 A B)
+    type-polynomial-endofunctor' A B (Enriched-Directed-Tree l3 l3 A B)
   pr1 (structure-coalgebra-Enriched-Directed-Tree T) =
     shape-root-Enriched-Directed-Tree A B T
   pr2 (structure-coalgebra-Enriched-Directed-Tree T) =
diff --git a/src/trees/coalgebras-polynomial-endofunctors.lagda.md b/src/trees/coalgebras-polynomial-endofunctors.lagda.md
index 85f9f915c3..ab0f5c5e7b 100644
--- a/src/trees/coalgebras-polynomial-endofunctors.lagda.md
+++ b/src/trees/coalgebras-polynomial-endofunctors.lagda.md
@@ -33,7 +33,7 @@ module _
 
   coalgebra-polynomial-endofunctor : UU (l1 ⊔ l2 ⊔ lsuc l)
   coalgebra-polynomial-endofunctor =
-    Σ (UU l) (λ X → X → type-polynomial-endofunctor A B X)
+    Σ (UU l) (λ X → X → type-polynomial-endofunctor' A B X)
 
 module _
   {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2}
@@ -45,7 +45,7 @@ module _
 
   structure-coalgebra-polynomial-endofunctor :
     type-coalgebra-polynomial-endofunctor →
-    type-polynomial-endofunctor A B type-coalgebra-polynomial-endofunctor
+    type-polynomial-endofunctor' A B type-coalgebra-polynomial-endofunctor
   structure-coalgebra-polynomial-endofunctor = pr2 X
 
   shape-coalgebra-polynomial-endofunctor :
diff --git a/src/trees/morphisms-algebras-polynomial-endofunctors.lagda.md b/src/trees/morphisms-algebras-polynomial-endofunctors.lagda.md
index d40f1944ee..78d7ec75ab 100644
--- a/src/trees/morphisms-algebras-polynomial-endofunctors.lagda.md
+++ b/src/trees/morphisms-algebras-polynomial-endofunctors.lagda.md
@@ -61,7 +61,7 @@ hom-algebra-polynomial-endofunctor {A = A} {B} X Y =
     ( λ f →
       ( f ∘ (structure-algebra-polynomial-endofunctor X)) ~
       ( ( structure-algebra-polynomial-endofunctor Y) ∘
-        ( map-polynomial-endofunctor A B f)))
+        ( map-polynomial-endofunctor' A B f)))
 
 map-hom-algebra-polynomial-endofunctor :
   {l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2}
@@ -80,7 +80,7 @@ structure-hom-algebra-polynomial-endofunctor :
   ( ( map-hom-algebra-polynomial-endofunctor X Y f) ∘
     ( structure-algebra-polynomial-endofunctor X)) ~
   ( ( structure-algebra-polynomial-endofunctor Y) ∘
-    ( map-polynomial-endofunctor A B
+    ( map-polynomial-endofunctor' A B
       ( map-hom-algebra-polynomial-endofunctor X Y f)))
 structure-hom-algebra-polynomial-endofunctor X Y f = pr2 f
 ```
@@ -105,7 +105,7 @@ module _
       ( λ H →
         ( ( structure-hom-algebra-polynomial-endofunctor X Y f) ∙h
           ( ( structure-algebra-polynomial-endofunctor Y) ·l
-            ( htpy-polynomial-endofunctor A B H))) ~
+            ( htpy-polynomial-endofunctor' A B H))) ~
         ( ( H ·r structure-algebra-polynomial-endofunctor X) ∙h
           ( structure-hom-algebra-polynomial-endofunctor X Y g)))
 
@@ -118,10 +118,10 @@ module _
         concat
           ( structure-hom-algebra-polynomial-endofunctor X Y f z)
           ( structure-algebra-polynomial-endofunctor Y
-            ( map-polynomial-endofunctor A B
+            ( map-polynomial-endofunctor' A B
               ( map-hom-algebra-polynomial-endofunctor X Y f) z))
           ( ap (structure-algebra-polynomial-endofunctor Y) t))
-      ( coh-refl-htpy-polynomial-endofunctor A B
+      ( coh-refl-htpy-polynomial-endofunctor' A B
         ( map-hom-algebra-polynomial-endofunctor X Y f) z)) ∙
     ( right-unit)
 
@@ -139,7 +139,7 @@ module _
       ( pair (map-hom-algebra-polynomial-endofunctor X Y f) refl-htpy)
       ( is-contr-equiv'
         ( Σ ( ( (pr1 f) ∘ pr2 X) ~
-              ( pr2 Y ∘ map-polynomial-endofunctor A B (pr1 f)))
+              ( pr2 Y ∘ map-polynomial-endofunctor' A B (pr1 f)))
             ( λ H → (pr2 f) ~ H))
         ( equiv-tot
           ( λ H →
@@ -149,10 +149,10 @@ module _
                   ( concat
                     ( pr2 f x)
                     ( structure-algebra-polynomial-endofunctor Y
-                      ( map-polynomial-endofunctor A B (pr1 f) x)))
+                      ( map-polynomial-endofunctor' A B (pr1 f) x)))
                   ( ap
                     ( ap (pr2 Y))
-                    ( coh-refl-htpy-polynomial-endofunctor A B (pr1 f) x)))
+                    ( coh-refl-htpy-polynomial-endofunctor' A B (pr1 f) x)))
               ( H)) ∘e
             ( equiv-concat-htpy right-unit-htpy H)))
         ( is-torsorial-htpy (pr2 f)))
diff --git a/src/trees/morphisms-coalgebras-polynomial-endofunctors.lagda.md b/src/trees/morphisms-coalgebras-polynomial-endofunctors.lagda.md
index f6771ba67b..0f883d63c1 100644
--- a/src/trees/morphisms-coalgebras-polynomial-endofunctors.lagda.md
+++ b/src/trees/morphisms-coalgebras-polynomial-endofunctors.lagda.md
@@ -62,7 +62,7 @@ hom-coalgebra-polynomial-endofunctor {A = A} {B} X Y =
       ( coherence-square-maps f
           ( structure-coalgebra-polynomial-endofunctor X)
           ( structure-coalgebra-polynomial-endofunctor Y)
-          ( map-polynomial-endofunctor A B f)))
+          ( map-polynomial-endofunctor' A B f)))
 
 map-hom-coalgebra-polynomial-endofunctor :
   {l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2}
@@ -82,7 +82,7 @@ structure-hom-coalgebra-polynomial-endofunctor :
     ( map-hom-coalgebra-polynomial-endofunctor X Y f)
     ( structure-coalgebra-polynomial-endofunctor X)
     ( structure-coalgebra-polynomial-endofunctor Y)
-    ( map-polynomial-endofunctor A B
+    ( map-polynomial-endofunctor' A B
       ( map-hom-coalgebra-polynomial-endofunctor X Y f))
 structure-hom-coalgebra-polynomial-endofunctor X Y f = pr2 f
 ```
@@ -107,7 +107,7 @@ module _
       ( λ H →
         ( ( structure-hom-coalgebra-polynomial-endofunctor X Y f) ∙h
           ( structure-coalgebra-polynomial-endofunctor Y ·l H)) ~
-        ( ( ( htpy-polynomial-endofunctor A B H) ·r
+        ( ( ( htpy-polynomial-endofunctor' A B H) ·r
             ( structure-coalgebra-polynomial-endofunctor X)) ∙h
           ( structure-hom-coalgebra-polynomial-endofunctor X Y g)))
 
@@ -119,11 +119,11 @@ module _
     ( inv
       ( ap
         ( concat'
-          ( map-polynomial-endofunctor A B
+          ( map-polynomial-endofunctor' A B
             ( map-hom-coalgebra-polynomial-endofunctor X Y f)
             ( structure-coalgebra-polynomial-endofunctor X z))
           ( structure-hom-coalgebra-polynomial-endofunctor X Y f z))
-        ( coh-refl-htpy-polynomial-endofunctor A B
+        ( coh-refl-htpy-polynomial-endofunctor' A B
           ( map-hom-coalgebra-polynomial-endofunctor X Y f)
           ( structure-coalgebra-polynomial-endofunctor X z))))
 
@@ -144,7 +144,7 @@ module _
               ( map-hom-coalgebra-polynomial-endofunctor X Y f)
               ( structure-coalgebra-polynomial-endofunctor X)
               ( structure-coalgebra-polynomial-endofunctor Y)
-              ( map-polynomial-endofunctor A B
+              ( map-polynomial-endofunctor' A B
                 ( map-hom-coalgebra-polynomial-endofunctor X Y f)))
             ( λ G →
               ( ( structure-hom-coalgebra-polynomial-endofunctor X Y f) ∙h
@@ -158,12 +158,12 @@ module _
               ( λ x →
                 ap
                   ( concat'
-                    ( ( map-polynomial-endofunctor A B
+                    ( ( map-polynomial-endofunctor' A B
                         ( map-hom-coalgebra-polynomial-endofunctor X Y f)
                         ( structure-coalgebra-polynomial-endofunctor X x)))
                     (G x))
                   ( inv
-                    ( coh-refl-htpy-polynomial-endofunctor A B
+                    ( coh-refl-htpy-polynomial-endofunctor' A B
                       ( map-hom-coalgebra-polynomial-endofunctor X Y f)
                       ( structure-coalgebra-polynomial-endofunctor X x))))))
         ( is-torsorial-htpy
diff --git a/src/trees/polynomial-endofunctors.lagda.md b/src/trees/polynomial-endofunctors.lagda.md
index e43a9d39d7..5479e449a0 100644
--- a/src/trees/polynomial-endofunctors.lagda.md
+++ b/src/trees/polynomial-endofunctors.lagda.md
@@ -31,7 +31,8 @@ open import foundation-core.torsorial-type-families
 ## Idea
 
 Given a type `A` [equipped](foundation.structure.md) with a type family `B` over
-`A`, the {{#concept "polynomial endofunctor"}} `𝑃 A B` is defined by
+`A`, the {{#concept "polynomial endofunctor" Agda=polynomial-endofunctor}}
+`𝑃 A B` is defined by
 
 ```text
   X ↦ Σ (x : A), (B x → X)
@@ -43,13 +44,38 @@ Polynomial endofunctors are important in the study of
 
 ## Definitions
 
+### The type of polynomial endofunctors
+
+```agda
+polynomial-endofunctor : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)
+polynomial-endofunctor l1 l2 = Σ (UU l1) (λ A → (A → UU l2))
+
+module _
+  {l1 l2 : Level} (𝑃 : polynomial-endofunctor l1 l2)
+  where
+
+  shapes-polynomial-endofunctor : UU l1
+  shapes-polynomial-endofunctor = pr1 𝑃
+
+  positions-polynomial-endofunctor : shapes-polynomial-endofunctor → UU l2
+  positions-polynomial-endofunctor = pr2 𝑃
+
+make-polynomial-endofunctor :
+  {l1 l2 : Level} {A : UU l1} → (A → UU l2) → polynomial-endofunctor l1 l2
+make-polynomial-endofunctor B = (_ , B)
+```
+
 ### The action on types of a polynomial endofunctor
 
 ```agda
-type-polynomial-endofunctor :
+type-polynomial-endofunctor' :
   {l1 l2 l3 : Level} (A : UU l1) (B : A → UU l2) (X : UU l3) →
   UU (l1 ⊔ l2 ⊔ l3)
-type-polynomial-endofunctor A B X = Σ A (λ x → B x → X)
+type-polynomial-endofunctor' A B X = Σ A (λ x → B x → X)
+
+type-polynomial-endofunctor :
+  {l1 l2 l3 : Level} → polynomial-endofunctor l1 l2 → UU l3 → UU (l1 ⊔ l2 ⊔ l3)
+type-polynomial-endofunctor (A , B) = type-polynomial-endofunctor' A B
 ```
 
 ### The identity type of `type-polynomial-endofunctor`
@@ -60,23 +86,23 @@ module _
   where
 
   Eq-type-polynomial-endofunctor :
-    (x y : type-polynomial-endofunctor A B X) → UU (l1 ⊔ l2 ⊔ l3)
+    (x y : type-polynomial-endofunctor' A B X) → UU (l1 ⊔ l2 ⊔ l3)
   Eq-type-polynomial-endofunctor x y =
     Σ (pr1 x ＝ pr1 y) (λ p → coherence-triangle-maps (pr2 x) (pr2 y) (tr B p))
 
   refl-Eq-type-polynomial-endofunctor :
-    (x : type-polynomial-endofunctor A B X) →
+    (x : type-polynomial-endofunctor' A B X) →
     Eq-type-polynomial-endofunctor x x
   refl-Eq-type-polynomial-endofunctor (x , α) = (refl , refl-htpy)
 
   Eq-eq-type-polynomial-endofunctor :
-    (x y : type-polynomial-endofunctor A B X) →
+    (x y : type-polynomial-endofunctor' A B X) →
     x ＝ y → Eq-type-polynomial-endofunctor x y
   Eq-eq-type-polynomial-endofunctor x .x refl =
     refl-Eq-type-polynomial-endofunctor x
 
   is-torsorial-Eq-type-polynomial-endofunctor :
-    (x : type-polynomial-endofunctor A B X) →
+    (x : type-polynomial-endofunctor' A B X) →
     is-torsorial (Eq-type-polynomial-endofunctor x)
   is-torsorial-Eq-type-polynomial-endofunctor (x , α) =
     is-torsorial-Eq-structure
@@ -85,7 +111,7 @@ module _
       ( is-torsorial-htpy α)
 
   is-equiv-Eq-eq-type-polynomial-endofunctor :
-    (x y : type-polynomial-endofunctor A B X) →
+    (x y : type-polynomial-endofunctor' A B X) →
     is-equiv (Eq-eq-type-polynomial-endofunctor x y)
   is-equiv-Eq-eq-type-polynomial-endofunctor x =
     fundamental-theorem-id
@@ -93,13 +119,13 @@ module _
       ( Eq-eq-type-polynomial-endofunctor x)
 
   eq-Eq-type-polynomial-endofunctor :
-    (x y : type-polynomial-endofunctor A B X) →
+    (x y : type-polynomial-endofunctor' A B X) →
     Eq-type-polynomial-endofunctor x y → x ＝ y
   eq-Eq-type-polynomial-endofunctor x y =
     map-inv-is-equiv (is-equiv-Eq-eq-type-polynomial-endofunctor x y)
 
   is-retraction-eq-Eq-type-polynomial-endofunctor :
-    (x y : type-polynomial-endofunctor A B X) →
+    (x y : type-polynomial-endofunctor' A B X) →
     is-retraction
       ( Eq-eq-type-polynomial-endofunctor x y)
       ( eq-Eq-type-polynomial-endofunctor x y)
@@ -108,7 +134,7 @@ module _
       ( is-equiv-Eq-eq-type-polynomial-endofunctor x y)
 
   coh-refl-eq-Eq-type-polynomial-endofunctor :
-    (x : type-polynomial-endofunctor A B X) →
+    (x : type-polynomial-endofunctor' A B X) →
     ( eq-Eq-type-polynomial-endofunctor x x
       ( refl-Eq-type-polynomial-endofunctor x)) ＝ refl
   coh-refl-eq-Eq-type-polynomial-endofunctor x =
@@ -118,33 +144,52 @@ module _
 ### The action on maps of the polynomial endofunctor
 
 ```agda
-map-polynomial-endofunctor :
+map-polynomial-endofunctor' :
   {l1 l2 l3 l4 : Level} (A : UU l1) (B : A → UU l2) {X : UU l3} {Y : UU l4}
   (f : X → Y) →
-  type-polynomial-endofunctor A B X → type-polynomial-endofunctor A B Y
-map-polynomial-endofunctor A B f = tot (λ x α → f ∘ α)
+  type-polynomial-endofunctor' A B X → type-polynomial-endofunctor' A B Y
+map-polynomial-endofunctor' A B f = tot (λ x α → f ∘ α)
+
+map-polynomial-endofunctor :
+  {l1 l2 l3 l4 : Level} (𝑃 : polynomial-endofunctor l1 l2)
+  {X : UU l3} {Y : UU l4} (f : X → Y) →
+  type-polynomial-endofunctor 𝑃 X → type-polynomial-endofunctor 𝑃 Y
+map-polynomial-endofunctor (A , B) = map-polynomial-endofunctor' A B
 ```
 
 ### The action on homotopies of the polynomial endofunctor
 
 ```agda
-htpy-polynomial-endofunctor :
+htpy-polynomial-endofunctor' :
   {l1 l2 l3 l4 : Level} (A : UU l1) (B : A → UU l2) {X : UU l3} {Y : UU l4}
   {f g : X → Y} →
-  f ~ g → map-polynomial-endofunctor A B f ~ map-polynomial-endofunctor A B g
-htpy-polynomial-endofunctor A B {f = f} {g} H (x , α) =
+  f ~ g → map-polynomial-endofunctor' A B f ~ map-polynomial-endofunctor' A B g
+htpy-polynomial-endofunctor' A B {f = f} {g} H (x , α) =
   eq-Eq-type-polynomial-endofunctor
-    ( map-polynomial-endofunctor A B f (x , α))
-    ( map-polynomial-endofunctor A B g (x , α))
+    ( map-polynomial-endofunctor' A B f (x , α))
+    ( map-polynomial-endofunctor' A B g (x , α))
     ( refl , H ·r α)
 
-coh-refl-htpy-polynomial-endofunctor :
-  {l1 l2 l3 l4 : Level} (A : UU l1) (B : A → UU l2) {X : UU l3} {Y : UU l4}
-  (f : X → Y) →
-  htpy-polynomial-endofunctor A B (refl-htpy {f = f}) ~ refl-htpy
-coh-refl-htpy-polynomial-endofunctor A B f (x , α) =
+htpy-polynomial-endofunctor :
+  {l1 l2 l3 l4 : Level} (𝑃 : polynomial-endofunctor l1 l2)
+  {X : UU l3} {Y : UU l4} {f g : X → Y} →
+  f ~ g → map-polynomial-endofunctor 𝑃 f ~ map-polynomial-endofunctor 𝑃 g
+htpy-polynomial-endofunctor (A , B) = htpy-polynomial-endofunctor' A B
+
+coh-refl-htpy-polynomial-endofunctor' :
+  {l1 l2 l3 l4 : Level} (A : UU l1) (B : A → UU l2)
+  {X : UU l3} {Y : UU l4} (f : X → Y) →
+  htpy-polynomial-endofunctor' A B (refl-htpy' f) ~ refl-htpy
+coh-refl-htpy-polynomial-endofunctor' A B f (x , α) =
   coh-refl-eq-Eq-type-polynomial-endofunctor
-    ( map-polynomial-endofunctor A B f (x , α))
+    ( map-polynomial-endofunctor' A B f (x , α))
+
+coh-refl-htpy-polynomial-endofunctor :
+  {l1 l2 l3 l4 : Level} (𝑃 : polynomial-endofunctor l1 l2)
+  {X : UU l3} {Y : UU l4} (f : X → Y) →
+  htpy-polynomial-endofunctor 𝑃 (refl-htpy' f) ~ refl-htpy
+coh-refl-htpy-polynomial-endofunctor (A , B) =
+  coh-refl-htpy-polynomial-endofunctor' A B
 ```
 
 ## See also
diff --git a/src/trees/w-types.lagda.md b/src/trees/w-types.lagda.md
index cf2df81dea..6e6426257f 100644
--- a/src/trees/w-types.lagda.md
+++ b/src/trees/w-types.lagda.md
@@ -70,7 +70,7 @@ module _
 ```agda
 structure-𝕎-Alg :
   {l1 l2 : Level} {A : UU l1} {B : A → UU l2} →
-  type-polynomial-endofunctor A B (𝕎 A B) → 𝕎 A B
+  type-polynomial-endofunctor' A B (𝕎 A B) → 𝕎 A B
 structure-𝕎-Alg (pair x α) = tree-𝕎 x α
 
 𝕎-Alg :
@@ -202,7 +202,7 @@ module _
 ```agda
 map-inv-structure-𝕎-Alg :
   {l1 l2 : Level} {A : UU l1} {B : A → UU l2} →
-  𝕎 A B → type-polynomial-endofunctor A B (𝕎 A B)
+  𝕎 A B → type-polynomial-endofunctor' A B (𝕎 A B)
 map-inv-structure-𝕎-Alg (tree-𝕎 x α) = pair x α
 
 is-section-map-inv-structure-𝕎-Alg :
@@ -226,7 +226,7 @@ is-equiv-structure-𝕎-Alg =
 
 equiv-structure-𝕎-Alg :
   {l1 l2 : Level} {A : UU l1} {B : A → UU l2} →
-  type-polynomial-endofunctor A B (𝕎 A B) ≃ 𝕎 A B
+  type-polynomial-endofunctor' A B (𝕎 A B) ≃ 𝕎 A B
 equiv-structure-𝕎-Alg =
   pair structure-𝕎-Alg is-equiv-structure-𝕎-Alg
 
@@ -241,7 +241,7 @@ is-equiv-map-inv-structure-𝕎-Alg =
 
 inv-equiv-structure-𝕎-Alg :
   {l1 l2 : Level} {A : UU l1} {B : A → UU l2} →
-  𝕎 A B ≃ type-polynomial-endofunctor A B (𝕎 A B)
+  𝕎 A B ≃ type-polynomial-endofunctor' A B (𝕎 A B)
 inv-equiv-structure-𝕎-Alg =
   pair map-inv-structure-𝕎-Alg is-equiv-map-inv-structure-𝕎-Alg
 ```
@@ -262,7 +262,7 @@ structure-hom-𝕎-Alg :
   (X : algebra-polynomial-endofunctor l3 A B) →
   ( (map-hom-𝕎-Alg X) ∘ structure-𝕎-Alg) ~
   ( ( structure-algebra-polynomial-endofunctor X) ∘
-    ( map-polynomial-endofunctor A B (map-hom-𝕎-Alg X)))
+    ( map-polynomial-endofunctor' A B (map-hom-𝕎-Alg X)))
 structure-hom-𝕎-Alg X (pair x α) = refl
 
 hom-𝕎-Alg :
@@ -292,7 +292,7 @@ compute-structure-htpy-hom-𝕎-Alg :
   (H : map-hom-𝕎-Alg X ~ f) →
   ( ap
     ( structure-algebra-polynomial-endofunctor X)
-    ( htpy-polynomial-endofunctor A B H (pair x α))) ＝
+    ( htpy-polynomial-endofunctor' A B H (pair x α))) ＝
   ( ap
     ( λ t → structure-algebra-polynomial-endofunctor X (pair x t))
     ( htpy-postcomp (B x) H α))
@@ -302,13 +302,13 @@ compute-structure-htpy-hom-𝕎-Alg {A = A} {B} X x α =
     ( λ f H →
       ( ap
         ( structure-algebra-polynomial-endofunctor X)
-        ( htpy-polynomial-endofunctor A B H (pair x α))) ＝
+        ( htpy-polynomial-endofunctor' A B H (pair x α))) ＝
       ( ap
         ( λ t → structure-algebra-polynomial-endofunctor X (pair x t))
         ( htpy-postcomp (B x) H α)))
     ( ap
       ( ap (pr2 X))
-      ( coh-refl-htpy-polynomial-endofunctor A B
+      ( coh-refl-htpy-polynomial-endofunctor' A B
         ( map-hom-𝕎-Alg X)
         ( pair x α)) ∙
     ( inv
@@ -322,7 +322,7 @@ structure-htpy-hom-𝕎-Alg :
   (f : hom-algebra-polynomial-endofunctor (𝕎-Alg A B) X) →
   ( structure-hom-𝕎-Alg X ∙h
     ( ( structure-algebra-polynomial-endofunctor X) ·l
-      ( htpy-polynomial-endofunctor A B (htpy-htpy-hom-𝕎-Alg X f)))) ~
+      ( htpy-polynomial-endofunctor' A B (htpy-htpy-hom-𝕎-Alg X f)))) ~
   ( ( (htpy-htpy-hom-𝕎-Alg X f) ·r structure-𝕎-Alg {B = B}) ∙h
     ( structure-hom-algebra-polynomial-endofunctor (𝕎-Alg A B) X f))
 structure-htpy-hom-𝕎-Alg {A = A} {B} X (pair f μ-f) (pair x α) =
@@ -334,7 +334,7 @@ structure-htpy-hom-𝕎-Alg {A = A} {B} X (pair f μ-f) (pair x α) =
         ( ap
           ( λ t → pr2 X (pair x t))
           ( eq-htpy (htpy-htpy-hom-𝕎-Alg X (pair f μ-f) ·r α)))
-        ( pr2 X (map-polynomial-endofunctor A B f (pair x α))))
+        ( pr2 X (map-polynomial-endofunctor' A B f (pair x α))))
       ( inv (left-inv ( μ-f (pair x α)))))) ∙
   ( inv
     ( assoc

From 7a75548f14dc6f57bbece65869a7546bbaba99f8 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 30 Aug 2025 23:21:22 +0200
Subject: [PATCH 02/41] natural transformations of polynomial endofunctors

---
 ...unctoriality-dependent-pair-types.lagda.md |   1 -
 ...equality-dependent-function-types.lagda.md |   1 -
 src/trees.lagda.md                            |   1 +
 ...ormations-polynomial-endofunctors.lagda.md | 277 ++++++++++++++++++
 4 files changed, 278 insertions(+), 2 deletions(-)
 create mode 100644 src/trees/natural-transformations-polynomial-endofunctors.lagda.md

diff --git a/src/foundation-core/functoriality-dependent-pair-types.lagda.md b/src/foundation-core/functoriality-dependent-pair-types.lagda.md
index 30b3e05d74..35d6415cb5 100644
--- a/src/foundation-core/functoriality-dependent-pair-types.lagda.md
+++ b/src/foundation-core/functoriality-dependent-pair-types.lagda.md
@@ -586,7 +586,6 @@ module _
         ap (g a') (tr-ap pr1 (λ x _ → pr2 x) p a)) ＝
     ap (tot g) {a , x} {a' , x'} p
   inv-compute-ap-tot' p = inv (compute-ap-tot' p)
-
 ```
 
 ### Computing the action on identifications of the functorial action of Σ
diff --git a/src/foundation/equality-dependent-function-types.lagda.md b/src/foundation/equality-dependent-function-types.lagda.md
index 5a58c07765..f42a34a801 100644
--- a/src/foundation/equality-dependent-function-types.lagda.md
+++ b/src/foundation/equality-dependent-function-types.lagda.md
@@ -55,7 +55,6 @@ module _
         ( λ _ → equiv-explicit-implicit-Π))
       ( is-torsorial-Eq-Π)
 
-
   is-torsorial-Eq-implicit-Π' : is-torsorial (λ g → (x : A) → C x (g {x}))
   is-torsorial-Eq-implicit-Π' =
     is-contr-equiv
diff --git a/src/trees.lagda.md b/src/trees.lagda.md
index c2cfc02fbc..16b97923f4 100644
--- a/src/trees.lagda.md
+++ b/src/trees.lagda.md
@@ -45,6 +45,7 @@ open import trees.morphisms-enriched-directed-trees public
 open import trees.multiset-indexed-dependent-products-of-types public
 open import trees.multisets public
 open import trees.multivariable-polynomial-functors public
+open import trees.natural-transformations-polynomial-endofunctors public
 open import trees.planar-binary-trees public
 open import trees.plane-trees public
 open import trees.polynomial-endofunctors public
diff --git a/src/trees/natural-transformations-polynomial-endofunctors.lagda.md b/src/trees/natural-transformations-polynomial-endofunctors.lagda.md
new file mode 100644
index 0000000000..03425e401f
--- /dev/null
+++ b/src/trees/natural-transformations-polynomial-endofunctors.lagda.md
@@ -0,0 +1,277 @@
+# Natural transformations between polynomial endofunctors
+
+```agda
+module trees.natural-transformations-polynomial-endofunctors where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.action-on-identifications-binary-functions
+open import foundation.action-on-identifications-functions
+open import foundation.binary-homotopies
+open import foundation.commuting-squares-of-homotopies
+open import foundation.commuting-squares-of-maps
+open import foundation.contractible-types
+open import foundation.dependent-pair-types
+open import foundation.equality-dependent-function-types
+open import foundation.equivalences
+open import foundation.function-types
+open import foundation.functoriality-dependent-pair-types
+open import foundation.fundamental-theorem-of-identity-types
+open import foundation.homotopies
+open import foundation.homotopy-induction
+open import foundation.identity-types
+open import foundation.implicit-function-types
+open import foundation.precomposition-functions
+open import foundation.sections
+open import foundation.structure-identity-principle
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+open import foundation.whiskering-homotopies-composition
+
+open import foundation-core.equality-dependent-pair-types
+open import foundation-core.retractions
+open import foundation-core.torsorial-type-families
+
+open import trees.polynomial-endofunctors
+```
+
+</details>
+
+## Idea
+
+Given two [polynomial endofunctors](trees.polynomial-endofunctors.md)
+$𝑃 ≐ (A ◃ B)$ and $𝑄 ≐ (C ◃ D)$, a
+{{#concept "natural transformation" Disambiguation="of polynomial endofunctors of types" Agda=natural-transformation-polynomial-endofunctor}}
+$α$ from $𝑃$ to $𝑄$ is a family of maps $α : (X : Type) → 𝑃(X) → 𝑄(X)$ such that
+for every map of types $f : X → Y$, the following square commutes
+
+```text
+              𝑃(f)
+     𝑃(X) -----------> 𝑃(Y)
+       |                |
+       |                |
+  α(X) |                | α(Y)
+       |                |
+       ∨                ∨
+     𝑄(X) -----------> 𝑄(Y).
+              𝑄(f)
+```
+
+## Definitions
+
+### Natural transformations between polynomial endofunctors
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  where
+
+  coherence-natural-transformation-polynomial-endofunctor :
+    {l : Level} →
+    ( {X : UU l} →
+      type-polynomial-endofunctor 𝑃 X →
+      type-polynomial-endofunctor 𝑄 X) →
+    UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ lsuc l)
+  coherence-natural-transformation-polynomial-endofunctor {l} φ₀ =
+    {X Y : UU l} (f : X → Y) →
+    coherence-square-maps
+      ( map-polynomial-endofunctor 𝑃 f)
+      ( φ₀)
+      ( φ₀)
+      ( map-polynomial-endofunctor 𝑄 f)
+
+  natural-transformation-polynomial-endofunctor :
+    (l : Level) → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ lsuc l)
+  natural-transformation-polynomial-endofunctor l =
+    Σ ( {X : UU l} →
+        type-polynomial-endofunctor 𝑃 X →
+        type-polynomial-endofunctor 𝑄 X)
+      ( coherence-natural-transformation-polynomial-endofunctor)
+
+module _
+  {l1 l2 l3 l4 l5 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5)
+  where
+
+  type-natural-transformation-polynomial-endofunctor :
+    {X : UU l5} →
+    type-polynomial-endofunctor 𝑃 X →
+    type-polynomial-endofunctor 𝑄 X
+  type-natural-transformation-polynomial-endofunctor = pr1 α
+
+  naturality-natural-transformation-polynomial-endofunctor :
+    coherence-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( type-natural-transformation-polynomial-endofunctor)
+  naturality-natural-transformation-polynomial-endofunctor = pr2 α
+```
+
+### The identity natural transformation
+
+```agda
+id-natural-transformation-polynomial-endofunctor :
+  {l1 l2 l3 : Level} (𝑃 : polynomial-endofunctor l1 l2) →
+  natural-transformation-polynomial-endofunctor 𝑃 𝑃 l3
+pr1 (id-natural-transformation-polynomial-endofunctor 𝑃) = id
+pr2 (id-natural-transformation-polynomial-endofunctor 𝑃) f x = refl
+```
+
+## Properties
+
+### Characterizing equality of natural transformations
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  where
+
+  htpy-natural-transformation-polynomial-endofunctor :
+    (α β : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5) →
+    UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ lsuc l5)
+  htpy-natural-transformation-polynomial-endofunctor α β =
+    Σ ( (X : UU l5) →
+        type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α {X} ~
+        type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 β {X})
+      ( λ H →
+        (X Y : UU l5) (f : X → Y) →
+        coherence-square-homotopies
+          ( map-polynomial-endofunctor 𝑄 f ·l H X)
+          ( naturality-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α f)
+          ( naturality-natural-transformation-polynomial-endofunctor 𝑃 𝑄 β f)
+          ( H Y ·r map-polynomial-endofunctor 𝑃 f))
+
+  refl-htpy-natural-transformation-polynomial-endofunctor :
+    (α : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5) →
+    htpy-natural-transformation-polynomial-endofunctor α α
+  refl-htpy-natural-transformation-polynomial-endofunctor α =
+    ( (λ X x → refl) , (λ X Y f x → right-unit))
+
+  htpy-eq-natural-transformation-polynomial-endofunctor :
+    (α β : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5) →
+    (α ＝ β) → htpy-natural-transformation-polynomial-endofunctor α β
+  htpy-eq-natural-transformation-polynomial-endofunctor α .α refl =
+    refl-htpy-natural-transformation-polynomial-endofunctor α
+
+  is-torsorial-htpy-natural-transformation-polynomial-endofunctor :
+    (α : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5) →
+    is-torsorial (htpy-natural-transformation-polynomial-endofunctor α)
+  is-torsorial-htpy-natural-transformation-polynomial-endofunctor α =
+    is-torsorial-Eq-structure
+      ( is-torsorial-Eq-implicit-Π'
+        ( λ X →
+          is-torsorial-htpy
+            ( type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α))
+        )
+      ( type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α ,
+        ( λ _ _ → refl))
+      ( is-torsorial-Eq-implicit-Π'
+        ( λ X →
+          is-torsorial-Eq-implicit-Π'
+            ( λ Y →
+              is-torsorial-Eq-Π
+                ( λ f →
+                  is-torsorial-htpy
+                    ( ( naturality-natural-transformation-polynomial-endofunctor
+                          𝑃 𝑄 α f) ∙h
+                      ( refl-htpy))))))
+
+  is-equiv-htpy-eq-natural-transformation-polynomial-endofunctor :
+    (α β : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5) →
+    is-equiv (htpy-eq-natural-transformation-polynomial-endofunctor α β)
+  is-equiv-htpy-eq-natural-transformation-polynomial-endofunctor α =
+    fundamental-theorem-id
+      ( is-torsorial-htpy-natural-transformation-polynomial-endofunctor α)
+      ( htpy-eq-natural-transformation-polynomial-endofunctor α)
+
+  equiv-htpy-eq-natural-transformation-polynomial-endofunctor :
+    (α β : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5) →
+    (α ＝ β) ≃ htpy-natural-transformation-polynomial-endofunctor α β
+  equiv-htpy-eq-natural-transformation-polynomial-endofunctor α β =
+    ( htpy-eq-natural-transformation-polynomial-endofunctor α β ,
+      is-equiv-htpy-eq-natural-transformation-polynomial-endofunctor α β)
+
+  eq-htpy-natural-transformation-polynomial-endofunctor :
+    (α β : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5) →
+    htpy-natural-transformation-polynomial-endofunctor α β →
+    α ＝ β
+  eq-htpy-natural-transformation-polynomial-endofunctor α β =
+    map-inv-equiv
+      ( equiv-htpy-eq-natural-transformation-polynomial-endofunctor α β)
+```
+
+### Composition of natural transformations
+
+```agda
+comp-natural-transformation-polynomial-endofunctor :
+  {l1 l2 l3 l4 l5 l6 l : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (𝑅 : polynomial-endofunctor l5 l6) →
+  natural-transformation-polynomial-endofunctor 𝑄 𝑅 l →
+  natural-transformation-polynomial-endofunctor 𝑃 𝑄 l →
+  natural-transformation-polynomial-endofunctor 𝑃 𝑅 l
+comp-natural-transformation-polynomial-endofunctor 𝑃 𝑄 𝑅 (β₀ , β₁) (α₀ , α₁) =
+  ( ( β₀ ∘ α₀) ,
+    ( λ f →
+      pasting-vertical-coherence-square-maps
+        ( map-polynomial-endofunctor 𝑃 f)
+        ( α₀)
+        ( α₀)
+        ( map-polynomial-endofunctor 𝑄 f)
+        ( β₀)
+        ( β₀)
+        ( map-polynomial-endofunctor 𝑅 f)
+        ( α₁ f)
+        ( β₁ f)))
+```
+
+### Unit laws for composition of natural transformations
+
+```agda
+left-unit-law-comp-natural-transformation-polynomial-endofunctor :
+  {l1 l2 l3 l4 l : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4) →
+  (α : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l) →
+  ( comp-natural-transformation-polynomial-endofunctor 𝑃 𝑄 𝑄
+    ( id-natural-transformation-polynomial-endofunctor 𝑄)
+    ( α)) ＝
+  α
+left-unit-law-comp-natural-transformation-polynomial-endofunctor
+  𝑃 𝑄 α@(α₀ , α₁) =
+  eq-htpy-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+    ( comp-natural-transformation-polynomial-endofunctor 𝑃 𝑄 𝑄
+      ( id-natural-transformation-polynomial-endofunctor 𝑄)
+      ( α))
+    ( α)
+    ( ( λ X x → refl) , (λ X Y f x → right-unit ∙ ap-id (α₁ f x)))
+
+right-unit-law-comp-natural-transformation-polynomial-endofunctor :
+  {l1 l2 l3 l4 l : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4) →
+  (α : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l) →
+  ( comp-natural-transformation-polynomial-endofunctor 𝑃 𝑃 𝑄
+    ( α)
+    ( id-natural-transformation-polynomial-endofunctor 𝑃)) ＝
+  α
+right-unit-law-comp-natural-transformation-polynomial-endofunctor
+  𝑃 𝑄 α@(α₀ , α₁) =
+  eq-htpy-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+    ( comp-natural-transformation-polynomial-endofunctor 𝑃 𝑃 𝑄
+      ( α)
+      ( id-natural-transformation-polynomial-endofunctor 𝑃))
+    ( α)
+    ( ( λ X x → refl) , (λ X Y f x → right-unit ∙ right-unit))
+```
+
+## See also
+
+- [Morphisms of polynomial endofunctors](trees.morphisms-polynomial-endofunctors.md)

From 57afb14597bf8b4820376b4f7506a51f6c7fe524 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sun, 31 Aug 2025 19:25:57 +0200
Subject: [PATCH 03/41] edits natural transformations

---
 ...ormations-polynomial-endofunctors.lagda.md | 190 ++++++++++--------
 1 file changed, 110 insertions(+), 80 deletions(-)

diff --git a/src/trees/natural-transformations-polynomial-endofunctors.lagda.md b/src/trees/natural-transformations-polynomial-endofunctors.lagda.md
index 03425e401f..f1946f2c0e 100644
--- a/src/trees/natural-transformations-polynomial-endofunctors.lagda.md
+++ b/src/trees/natural-transformations-polynomial-endofunctors.lagda.md
@@ -20,9 +20,11 @@ open import foundation.function-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
+open import foundation.homotopies-morphisms-arrows
 open import foundation.homotopy-induction
 open import foundation.identity-types
 open import foundation.implicit-function-types
+open import foundation.morphisms-arrows
 open import foundation.precomposition-functions
 open import foundation.sections
 open import foundation.structure-identity-principle
@@ -48,15 +50,15 @@ $α$ from $𝑃$ to $𝑄$ is a family of maps $α : (X : Type) → 𝑃(X) →
 for every map of types $f : X → Y$, the following square commutes
 
 ```text
-              𝑃(f)
-     𝑃(X) -----------> 𝑃(Y)
+              α(X)
+     𝑃(X) -----------> 𝑄(X)
        |                |
        |                |
-  α(X) |                | α(Y)
+  𝑃(f) |                | 𝑄(f)
        |                |
        ∨                ∨
-     𝑄(X) -----------> 𝑄(Y).
-              𝑄(f)
+     𝑃(Y) -----------> 𝑄(Y).
+              α(Y)
 ```
 
 ## Definitions
@@ -64,39 +66,38 @@ for every map of types $f : X → Y$, the following square commutes
 ### Natural transformations between polynomial endofunctors
 
 ```agda
-module _
-  {l1 l2 l3 l4 : Level}
+coherence-natural-transformation-polynomial-endofunctor :
+  {l1 l2 l3 l4 l : Level}
   (𝑃 : polynomial-endofunctor l1 l2)
-  (𝑄 : polynomial-endofunctor l3 l4)
-  where
-
-  coherence-natural-transformation-polynomial-endofunctor :
-    {l : Level} →
-    ( {X : UU l} →
-      type-polynomial-endofunctor 𝑃 X →
-      type-polynomial-endofunctor 𝑄 X) →
+  (𝑄 : polynomial-endofunctor l3 l4) →
+  ( {X : UU l} →
+    type-polynomial-endofunctor 𝑃 X →
+    type-polynomial-endofunctor 𝑄 X) →
+  UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ lsuc l)
+coherence-natural-transformation-polynomial-endofunctor {l = l} 𝑃 𝑄 α₀ =
+  {X Y : UU l} (f : X → Y) →
+  coherence-square-maps
+    ( α₀)
+    ( map-polynomial-endofunctor 𝑃 f)
+    ( map-polynomial-endofunctor 𝑄 f)
+  ( α₀)
+
+natural-transformation-polynomial-endofunctor :
+  {l1 l2 l3 l4 : Level} (l : Level) →
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4) →
     UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ lsuc l)
-  coherence-natural-transformation-polynomial-endofunctor {l} φ₀ =
-    {X Y : UU l} (f : X → Y) →
-    coherence-square-maps
-      ( map-polynomial-endofunctor 𝑃 f)
-      ( φ₀)
-      ( φ₀)
-      ( map-polynomial-endofunctor 𝑄 f)
-
-  natural-transformation-polynomial-endofunctor :
-    (l : Level) → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ lsuc l)
-  natural-transformation-polynomial-endofunctor l =
-    Σ ( {X : UU l} →
-        type-polynomial-endofunctor 𝑃 X →
-        type-polynomial-endofunctor 𝑄 X)
-      ( coherence-natural-transformation-polynomial-endofunctor)
+natural-transformation-polynomial-endofunctor l 𝑃 𝑄 =
+  Σ ( {X : UU l} →
+      type-polynomial-endofunctor 𝑃 X →
+      type-polynomial-endofunctor 𝑄 X)
+    ( coherence-natural-transformation-polynomial-endofunctor 𝑃 𝑄)
 
 module _
   {l1 l2 l3 l4 l5 : Level}
   (𝑃 : polynomial-endofunctor l1 l2)
   (𝑄 : polynomial-endofunctor l3 l4)
-  (α : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5)
+  (α : natural-transformation-polynomial-endofunctor l5 𝑃 𝑄)
   where
 
   type-natural-transformation-polynomial-endofunctor :
@@ -111,12 +112,26 @@ module _
   naturality-natural-transformation-polynomial-endofunctor = pr2 α
 ```
 
+### The associated family of morphisms of arrows
+
+```agda
+hom-arrow-natural-transformation-polynomial-endofunctor :
+  {l1 l2 l3 l4 l5 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : natural-transformation-polynomial-endofunctor l5 𝑃 𝑄)
+  {X Y : UU l5} (f : X → Y) →
+  hom-arrow (map-polynomial-endofunctor 𝑃 f) (map-polynomial-endofunctor 𝑄 f)
+hom-arrow-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α@(α₀ , α₁) f =
+  ( α₀ , α₀ , α₁ f)
+```
+
 ### The identity natural transformation
 
 ```agda
 id-natural-transformation-polynomial-endofunctor :
   {l1 l2 l3 : Level} (𝑃 : polynomial-endofunctor l1 l2) →
-  natural-transformation-polynomial-endofunctor 𝑃 𝑃 l3
+  natural-transformation-polynomial-endofunctor l3 𝑃 𝑃
 pr1 (id-natural-transformation-polynomial-endofunctor 𝑃) = id
 pr2 (id-natural-transformation-polynomial-endofunctor 𝑃) f x = refl
 ```
@@ -133,7 +148,7 @@ module _
   where
 
   htpy-natural-transformation-polynomial-endofunctor :
-    (α β : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5) →
+    (α β : natural-transformation-polynomial-endofunctor l5 𝑃 𝑄) →
     UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ lsuc l5)
   htpy-natural-transformation-polynomial-endofunctor α β =
     Σ ( (X : UU l5) →
@@ -142,33 +157,32 @@ module _
       ( λ H →
         (X Y : UU l5) (f : X → Y) →
         coherence-square-homotopies
-          ( map-polynomial-endofunctor 𝑄 f ·l H X)
           ( naturality-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α f)
-          ( naturality-natural-transformation-polynomial-endofunctor 𝑃 𝑄 β f)
-          ( H Y ·r map-polynomial-endofunctor 𝑃 f))
+          ( H Y ·r map-polynomial-endofunctor 𝑃 f)
+          ( map-polynomial-endofunctor 𝑄 f ·l H X)
+          ( naturality-natural-transformation-polynomial-endofunctor 𝑃 𝑄 β f))
 
   refl-htpy-natural-transformation-polynomial-endofunctor :
-    (α : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5) →
+    (α : natural-transformation-polynomial-endofunctor l5 𝑃 𝑄) →
     htpy-natural-transformation-polynomial-endofunctor α α
   refl-htpy-natural-transformation-polynomial-endofunctor α =
-    ( (λ X x → refl) , (λ X Y f x → right-unit))
+    ( (λ X x → refl) , (λ X Y f x → inv right-unit))
 
   htpy-eq-natural-transformation-polynomial-endofunctor :
-    (α β : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5) →
+    (α β : natural-transformation-polynomial-endofunctor l5 𝑃 𝑄) →
     (α ＝ β) → htpy-natural-transformation-polynomial-endofunctor α β
   htpy-eq-natural-transformation-polynomial-endofunctor α .α refl =
     refl-htpy-natural-transformation-polynomial-endofunctor α
 
   is-torsorial-htpy-natural-transformation-polynomial-endofunctor :
-    (α : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5) →
+    (α : natural-transformation-polynomial-endofunctor l5 𝑃 𝑄) →
     is-torsorial (htpy-natural-transformation-polynomial-endofunctor α)
   is-torsorial-htpy-natural-transformation-polynomial-endofunctor α =
     is-torsorial-Eq-structure
       ( is-torsorial-Eq-implicit-Π'
         ( λ X →
           is-torsorial-htpy
-            ( type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α))
-        )
+            ( type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α)))
       ( type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α ,
         ( λ _ _ → refl))
       ( is-torsorial-Eq-implicit-Π'
@@ -177,13 +191,13 @@ module _
             ( λ Y →
               is-torsorial-Eq-Π
                 ( λ f →
-                  is-torsorial-htpy
+                  is-torsorial-htpy'
                     ( ( naturality-natural-transformation-polynomial-endofunctor
                           𝑃 𝑄 α f) ∙h
                       ( refl-htpy))))))
 
   is-equiv-htpy-eq-natural-transformation-polynomial-endofunctor :
-    (α β : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5) →
+    (α β : natural-transformation-polynomial-endofunctor l5 𝑃 𝑄) →
     is-equiv (htpy-eq-natural-transformation-polynomial-endofunctor α β)
   is-equiv-htpy-eq-natural-transformation-polynomial-endofunctor α =
     fundamental-theorem-id
@@ -191,14 +205,14 @@ module _
       ( htpy-eq-natural-transformation-polynomial-endofunctor α)
 
   equiv-htpy-eq-natural-transformation-polynomial-endofunctor :
-    (α β : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5) →
+    (α β : natural-transformation-polynomial-endofunctor l5 𝑃 𝑄) →
     (α ＝ β) ≃ htpy-natural-transformation-polynomial-endofunctor α β
   equiv-htpy-eq-natural-transformation-polynomial-endofunctor α β =
     ( htpy-eq-natural-transformation-polynomial-endofunctor α β ,
       is-equiv-htpy-eq-natural-transformation-polynomial-endofunctor α β)
 
   eq-htpy-natural-transformation-polynomial-endofunctor :
-    (α β : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l5) →
+    (α β : natural-transformation-polynomial-endofunctor l5 𝑃 𝑄) →
     htpy-natural-transformation-polynomial-endofunctor α β →
     α ＝ β
   eq-htpy-natural-transformation-polynomial-endofunctor α β =
@@ -214,20 +228,20 @@ comp-natural-transformation-polynomial-endofunctor :
   (𝑃 : polynomial-endofunctor l1 l2)
   (𝑄 : polynomial-endofunctor l3 l4)
   (𝑅 : polynomial-endofunctor l5 l6) →
-  natural-transformation-polynomial-endofunctor 𝑄 𝑅 l →
-  natural-transformation-polynomial-endofunctor 𝑃 𝑄 l →
-  natural-transformation-polynomial-endofunctor 𝑃 𝑅 l
+  natural-transformation-polynomial-endofunctor l 𝑄 𝑅 →
+  natural-transformation-polynomial-endofunctor l 𝑃 𝑄 →
+  natural-transformation-polynomial-endofunctor l 𝑃 𝑅
 comp-natural-transformation-polynomial-endofunctor 𝑃 𝑄 𝑅 (β₀ , β₁) (α₀ , α₁) =
   ( ( β₀ ∘ α₀) ,
     ( λ f →
-      pasting-vertical-coherence-square-maps
-        ( map-polynomial-endofunctor 𝑃 f)
-        ( α₀)
+      pasting-horizontal-coherence-square-maps
         ( α₀)
-        ( map-polynomial-endofunctor 𝑄 f)
-        ( β₀)
         ( β₀)
+        ( map-polynomial-endofunctor 𝑃 f)
+        ( map-polynomial-endofunctor 𝑄 f)
         ( map-polynomial-endofunctor 𝑅 f)
+        ( α₀)
+        ( β₀)
         ( α₁ f)
         ( β₁ f)))
 ```
@@ -235,41 +249,57 @@ comp-natural-transformation-polynomial-endofunctor 𝑃 𝑄 𝑅 (β₀ , β₁
 ### Unit laws for composition of natural transformations
 
 ```agda
-left-unit-law-comp-natural-transformation-polynomial-endofunctor :
+module _
   {l1 l2 l3 l4 l : Level}
   (𝑃 : polynomial-endofunctor l1 l2)
-  (𝑄 : polynomial-endofunctor l3 l4) →
-  (α : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l) →
-  ( comp-natural-transformation-polynomial-endofunctor 𝑃 𝑄 𝑄
-    ( id-natural-transformation-polynomial-endofunctor 𝑄)
-    ( α)) ＝
-  α
-left-unit-law-comp-natural-transformation-polynomial-endofunctor
-  𝑃 𝑄 α@(α₀ , α₁) =
-  eq-htpy-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α@(α₀ , α₁) : natural-transformation-polynomial-endofunctor l 𝑃 𝑄)
+  where
+
+  htpy-left-unit-law-comp-natural-transformation-polynomial-endofunctor :
+    htpy-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( comp-natural-transformation-polynomial-endofunctor 𝑃 𝑄 𝑄
+        ( id-natural-transformation-polynomial-endofunctor 𝑄)
+        ( α))
+      ( α)
+  htpy-left-unit-law-comp-natural-transformation-polynomial-endofunctor =
+      ( ( λ X x → refl) ,
+        ( λ X Y f x → inv (right-unit ∙ right-unit ∙ ap-id (α₁ f x))))
+
+  left-unit-law-comp-natural-transformation-polynomial-endofunctor :
     ( comp-natural-transformation-polynomial-endofunctor 𝑃 𝑄 𝑄
       ( id-natural-transformation-polynomial-endofunctor 𝑄)
-      ( α))
+      ( α)) ＝
     ( α)
-    ( ( λ X x → refl) , (λ X Y f x → right-unit ∙ ap-id (α₁ f x)))
+  left-unit-law-comp-natural-transformation-polynomial-endofunctor =
+    eq-htpy-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( comp-natural-transformation-polynomial-endofunctor 𝑃 𝑄 𝑄
+        ( id-natural-transformation-polynomial-endofunctor 𝑄)
+        ( α))
+      ( α)
+      ( htpy-left-unit-law-comp-natural-transformation-polynomial-endofunctor)
 
-right-unit-law-comp-natural-transformation-polynomial-endofunctor :
-  {l1 l2 l3 l4 l : Level}
-  (𝑃 : polynomial-endofunctor l1 l2)
-  (𝑄 : polynomial-endofunctor l3 l4) →
-  (α : natural-transformation-polynomial-endofunctor 𝑃 𝑄 l) →
-  ( comp-natural-transformation-polynomial-endofunctor 𝑃 𝑃 𝑄
-    ( α)
-    ( id-natural-transformation-polynomial-endofunctor 𝑃)) ＝
-  α
-right-unit-law-comp-natural-transformation-polynomial-endofunctor
-  𝑃 𝑄 α@(α₀ , α₁) =
-  eq-htpy-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+  htpy-right-unit-law-comp-natural-transformation-polynomial-endofunctor :
+    htpy-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( comp-natural-transformation-polynomial-endofunctor 𝑃 𝑃 𝑄
+        ( α)
+        ( id-natural-transformation-polynomial-endofunctor 𝑃))
+      ( α)
+  htpy-right-unit-law-comp-natural-transformation-polynomial-endofunctor =
+      ( ( λ X x → refl) , (λ X Y f x → inv right-unit))
+
+  right-unit-law-comp-natural-transformation-polynomial-endofunctor :
     ( comp-natural-transformation-polynomial-endofunctor 𝑃 𝑃 𝑄
       ( α)
-      ( id-natural-transformation-polynomial-endofunctor 𝑃))
+      ( id-natural-transformation-polynomial-endofunctor 𝑃)) ＝
     ( α)
-    ( ( λ X x → refl) , (λ X Y f x → right-unit ∙ right-unit))
+  right-unit-law-comp-natural-transformation-polynomial-endofunctor =
+    eq-htpy-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( comp-natural-transformation-polynomial-endofunctor 𝑃 𝑃 𝑄
+        ( α)
+        ( id-natural-transformation-polynomial-endofunctor 𝑃))
+      ( α)
+      ( htpy-right-unit-law-comp-natural-transformation-polynomial-endofunctor)
 ```
 
 ## See also

From 43be92848d8b01b7ee0c83a6d22d86b872273f5e Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sun, 31 Aug 2025 19:26:04 +0200
Subject: [PATCH 04/41] define morphisms of polynomial endofunctors

---
 src/trees.lagda.md                            |   1 +
 ...morphisms-polynomial-endofunctors.lagda.md | 294 ++++++++++++++++++
 2 files changed, 295 insertions(+)
 create mode 100644 src/trees/morphisms-polynomial-endofunctors.lagda.md

diff --git a/src/trees.lagda.md b/src/trees.lagda.md
index 16b97923f4..034d2ebc0c 100644
--- a/src/trees.lagda.md
+++ b/src/trees.lagda.md
@@ -42,6 +42,7 @@ open import trees.morphisms-algebras-polynomial-endofunctors public
 open import trees.morphisms-coalgebras-polynomial-endofunctors public
 open import trees.morphisms-directed-trees public
 open import trees.morphisms-enriched-directed-trees public
+open import trees.morphisms-polynomial-endofunctors public
 open import trees.multiset-indexed-dependent-products-of-types public
 open import trees.multisets public
 open import trees.multivariable-polynomial-functors public
diff --git a/src/trees/morphisms-polynomial-endofunctors.lagda.md b/src/trees/morphisms-polynomial-endofunctors.lagda.md
new file mode 100644
index 0000000000..e2f0208da4
--- /dev/null
+++ b/src/trees/morphisms-polynomial-endofunctors.lagda.md
@@ -0,0 +1,294 @@
+# Morphisms of polynomial endofunctors
+
+```agda
+module trees.morphisms-polynomial-endofunctors where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.action-on-identifications-functions
+open import foundation.binary-homotopies
+open import foundation.commuting-squares-of-homotopies
+open import foundation.commuting-squares-of-maps
+open import foundation.contractible-types
+open import foundation.dependent-pair-types
+open import foundation.equality-dependent-function-types
+open import foundation.equivalences
+open import foundation.function-types
+open import foundation.functoriality-dependent-pair-types
+open import foundation.fundamental-theorem-of-identity-types
+open import foundation.homotopies
+open import foundation.homotopy-induction
+open import foundation.identity-types
+open import foundation.implicit-function-types
+open import foundation.precomposition-functions
+open import foundation.sections
+open import foundation.structure-identity-principle
+open import foundation.transport-along-identifications
+open import foundation.truncated-types
+open import foundation.truncation-levels
+open import foundation.universe-levels
+open import foundation.whiskering-homotopies-composition
+
+open import foundation-core.equality-dependent-pair-types
+open import foundation-core.retractions
+open import foundation-core.torsorial-type-families
+
+open import trees.natural-transformations-polynomial-endofunctors
+open import trees.polynomial-endofunctors
+```
+
+</details>
+
+## Idea
+
+Given two [polynomial endofunctors](trees.polynomial-endofunctors.md)
+$𝑃 ≐ (A ◃ B)$ and $𝑄 ≐ (C ◃ D)$, a
+{{#concept "morphism" Disambiguation="of polynomial endofunctors of types" Agda=hom-polynomial-endofunctor}}
+$α$ from $𝑃$ to $𝑄$ consists of a map $α₀ : A → C$ and a family of maps
+$$α₁ : (a : A) → D (α₀ a) → B a.$$
+
+## Definitions
+
+### Morphisms of polynomial endofunctors
+
+```agda
+hom-polynomial-endofunctor :
+  {l1 l2 l3 l4 : Level} →
+  polynomial-endofunctor l1 l2 →
+  polynomial-endofunctor l3 l4 → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
+hom-polynomial-endofunctor (A , B) (C , D) =
+  Σ (A → C) (λ α₀ → ((a : A) → D (α₀ a) → B a))
+
+module _
+  {l1 l2 l3 l4 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : hom-polynomial-endofunctor 𝑃 𝑄)
+  where
+
+  shapes-hom-polynomial-endofunctor :
+    shapes-polynomial-endofunctor 𝑃 → shapes-polynomial-endofunctor 𝑄
+  shapes-hom-polynomial-endofunctor = pr1 α
+
+  positions-hom-polynomial-endofunctor :
+    (a : shapes-polynomial-endofunctor 𝑃) →
+    positions-polynomial-endofunctor 𝑄 (shapes-hom-polynomial-endofunctor a) →
+    positions-polynomial-endofunctor 𝑃 a
+  positions-hom-polynomial-endofunctor = pr2 α
+
+  type-hom-polynomial-endofunctor :
+    {l3 : Level} {X : UU l3} →
+    type-polynomial-endofunctor 𝑃 X →
+    type-polynomial-endofunctor 𝑄 X
+  type-hom-polynomial-endofunctor {X = X} =
+    map-Σ
+      ( λ c → positions-polynomial-endofunctor 𝑄 c → X)
+      ( shapes-hom-polynomial-endofunctor)
+      ( λ a → precomp (positions-hom-polynomial-endofunctor a) X)
+```
+
+## Properties
+
+### Characterizing equality of morphisms
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  where
+
+  htpy-hom-polynomial-endofunctor :
+    (α β : hom-polynomial-endofunctor 𝑃 𝑄) → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
+  htpy-hom-polynomial-endofunctor α β =
+    Σ ( shapes-hom-polynomial-endofunctor 𝑃 𝑄 α ~
+        shapes-hom-polynomial-endofunctor 𝑃 𝑄 β)
+      ( λ H →
+        (a : shapes-polynomial-endofunctor 𝑃)
+        (d :
+          positions-polynomial-endofunctor 𝑄
+            ( shapes-hom-polynomial-endofunctor 𝑃 𝑄 α a)) →
+        positions-hom-polynomial-endofunctor 𝑃 𝑄 α a d ＝
+        positions-hom-polynomial-endofunctor 𝑃 𝑄 β a
+          ( tr (positions-polynomial-endofunctor 𝑄) (H a) d))
+
+  refl-htpy-hom-polynomial-endofunctor :
+    (α : hom-polynomial-endofunctor 𝑃 𝑄) → htpy-hom-polynomial-endofunctor α α
+  refl-htpy-hom-polynomial-endofunctor α = (refl-htpy , λ a d → refl)
+
+  htpy-eq-hom-polynomial-endofunctor :
+    (α β : hom-polynomial-endofunctor 𝑃 𝑄) →
+    (α ＝ β) → htpy-hom-polynomial-endofunctor α β
+  htpy-eq-hom-polynomial-endofunctor α .α refl =
+    refl-htpy-hom-polynomial-endofunctor α
+
+  is-torsorial-htpy-hom-polynomial-endofunctor :
+    (α : hom-polynomial-endofunctor 𝑃 𝑄) →
+    is-torsorial (htpy-hom-polynomial-endofunctor α)
+  is-torsorial-htpy-hom-polynomial-endofunctor α =
+    is-torsorial-Eq-structure
+      ( is-torsorial-htpy (shapes-hom-polynomial-endofunctor 𝑃 𝑄 α))
+      ( shapes-hom-polynomial-endofunctor 𝑃 𝑄 α , refl-htpy)
+      ( is-torsorial-binary-htpy (positions-hom-polynomial-endofunctor 𝑃 𝑄 α))
+
+  is-equiv-htpy-eq-hom-polynomial-endofunctor :
+    (α β : hom-polynomial-endofunctor 𝑃 𝑄) →
+    is-equiv (htpy-eq-hom-polynomial-endofunctor α β)
+  is-equiv-htpy-eq-hom-polynomial-endofunctor α =
+    fundamental-theorem-id
+      ( is-torsorial-htpy-hom-polynomial-endofunctor α)
+      ( htpy-eq-hom-polynomial-endofunctor α)
+
+  equiv-htpy-eq-hom-polynomial-endofunctor :
+    (α β : hom-polynomial-endofunctor 𝑃 𝑄) →
+    (α ＝ β) ≃ htpy-hom-polynomial-endofunctor α β
+  equiv-htpy-eq-hom-polynomial-endofunctor α β =
+    ( htpy-eq-hom-polynomial-endofunctor α β ,
+      is-equiv-htpy-eq-hom-polynomial-endofunctor α β)
+
+  eq-htpy-hom-polynomial-endofunctor :
+    (α β : hom-polynomial-endofunctor 𝑃 𝑄) →
+    htpy-hom-polynomial-endofunctor α β →
+    α ＝ β
+  eq-htpy-hom-polynomial-endofunctor α β =
+    map-inv-equiv (equiv-htpy-eq-hom-polynomial-endofunctor α β)
+```
+
+### Truncatedness of the type of morphisms
+
+If the shapes of $𝑄$ and the positions of $𝑃$ are $k$-truncated then the type of
+morphisms $𝑃 → 𝑄$ is $k$-truncated.
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  where
+
+  is-trunc-hom-polynomial-endofunctor :
+    (k : 𝕋) →
+    is-trunc k (shapes-polynomial-endofunctor 𝑄) →
+    ( (a : shapes-polynomial-endofunctor 𝑃) →
+      is-trunc k (positions-polynomial-endofunctor 𝑃 a)) →
+    is-trunc k (hom-polynomial-endofunctor 𝑃 𝑄)
+  is-trunc-hom-polynomial-endofunctor k hQ hP =
+    is-trunc-Σ
+      ( is-trunc-function-type k hQ)
+      ( λ f → is-trunc-Π k (λ a → is-trunc-function-type k (hP a)))
+```
+
+### Morphisms are natural transformations
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : hom-polynomial-endofunctor 𝑃 𝑄)
+  where
+
+  naturality-hom-polynomial-endofunctor :
+    {l5 l6 : Level} {X : UU l5} {Y : UU l6} (f : X → Y) →
+    coherence-square-maps
+      ( map-polynomial-endofunctor 𝑃 f)
+      ( type-hom-polynomial-endofunctor 𝑃 𝑄 α)
+      ( type-hom-polynomial-endofunctor 𝑃 𝑄 α)
+      ( map-polynomial-endofunctor 𝑄 f)
+  naturality-hom-polynomial-endofunctor f = refl-htpy
+
+  natural-transformation-hom-polynomial-endofunctor :
+    {l : Level} → natural-transformation-polynomial-endofunctor l 𝑃 𝑄
+  natural-transformation-hom-polynomial-endofunctor =
+    ( type-hom-polynomial-endofunctor 𝑃 𝑄 α ,
+      naturality-hom-polynomial-endofunctor)
+```
+
+### Natural transformations define morphisms
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : natural-transformation-polynomial-endofunctor l2 𝑃 𝑄)
+  where
+
+  shapes-natural-transformation-polynomial-endofunctor :
+    shapes-polynomial-endofunctor 𝑃 →
+    shapes-polynomial-endofunctor 𝑄
+  shapes-natural-transformation-polynomial-endofunctor a =
+    pr1 (type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α (a , id))
+
+  positions-natural-transformation-polynomial-endofunctor :
+    (a : shapes-polynomial-endofunctor 𝑃) →
+    positions-polynomial-endofunctor 𝑄
+      ( shapes-natural-transformation-polynomial-endofunctor a) →
+    positions-polynomial-endofunctor 𝑃 a
+  positions-natural-transformation-polynomial-endofunctor a =
+    pr2 (type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α (a , id))
+
+  hom-natural-transformation-polynomial-endofunctor :
+    hom-polynomial-endofunctor 𝑃 𝑄
+  hom-natural-transformation-polynomial-endofunctor =
+    ( shapes-natural-transformation-polynomial-endofunctor ,
+      positions-natural-transformation-polynomial-endofunctor)
+```
+
+### Equivalence between morphisms and natural transformations
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  where
+
+  is-retraction-hom-natural-transformation-polynomial-endofunctor :
+    is-retraction
+      ( λ α → natural-transformation-hom-polynomial-endofunctor 𝑃 𝑄 α {l2})
+      ( hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄)
+  is-retraction-hom-natural-transformation-polynomial-endofunctor α = refl
+
+  is-section-type-hom-natural-transformation-polynomial-endofunctor :
+    (α : natural-transformation-polynomial-endofunctor l2 𝑃 𝑄)
+    (X : UU l2) →
+    type-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( natural-transformation-hom-polynomial-endofunctor 𝑃 𝑄
+        ( hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α)) ~
+    type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α {X}
+  is-section-type-hom-natural-transformation-polynomial-endofunctor
+    α X (a , x) =
+    inv
+      ( naturality-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α x
+        ( a , id))
+```
+
+Finally, we need to show the following equality:
+
+$$
+  α₁ (f ∘ x) (a , id) = ap (𝑄 f) (α₁ x (a , id)) ∙ α₁ f (a , x)
+$$
+
+```text
+module _
+  {l : Level}
+  (𝑃 : polynomial-endofunctor l l)
+  (𝑄 : polynomial-endofunctor l l)
+  where
+
+  is-section-is-set-positions-hom-natural-transformation-polynomial-endofunctor :
+    is-section
+      ( λ α → natural-transformation-hom-polynomial-endofunctor 𝑃 𝑄 α {l})
+      ( hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄)
+  is-section-is-set-positions-hom-natural-transformation-polynomial-endofunctor α =
+    eq-htpy-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( natural-transformation-hom-polynomial-endofunctor 𝑃 𝑄
+        ( hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α))
+      ( α)
+      ( ( is-section-type-hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+          ( α)) ,
+        ?)
+```

From f8304fe3d17ec950ddf164ae346f6679ee162677 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sun, 31 Aug 2025 19:26:35 +0200
Subject: [PATCH 05/41] define cartesian natural transformations of polynomial
 endofunctors

---
 src/trees.lagda.md                            |   1 +
 ...ormations-polynomial-endofunctors.lagda.md | 228 ++++++++++++++++++
 2 files changed, 229 insertions(+)
 create mode 100644 src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md

diff --git a/src/trees.lagda.md b/src/trees.lagda.md
index 034d2ebc0c..a3fad6bbf4 100644
--- a/src/trees.lagda.md
+++ b/src/trees.lagda.md
@@ -14,6 +14,7 @@ open import trees.bases-directed-trees public
 open import trees.bases-enriched-directed-trees public
 open import trees.binary-w-types public
 open import trees.bounded-multisets public
+open import trees.cartesian-natural-transformations-polynomial-endofunctors public
 open import trees.coalgebra-of-directed-trees public
 open import trees.coalgebra-of-enriched-directed-trees public
 open import trees.coalgebras-polynomial-endofunctors public
diff --git a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
new file mode 100644
index 0000000000..107b903571
--- /dev/null
+++ b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
@@ -0,0 +1,228 @@
+# Cartesian natural transformations between polynomial endofunctors
+
+```agda
+module trees.cartesian-natural-transformations-polynomial-endofunctors where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.action-on-identifications-binary-functions
+open import foundation.action-on-identifications-functions
+open import foundation.binary-homotopies
+open import foundation.cartesian-morphisms-arrows
+open import foundation.commuting-squares-of-homotopies
+open import foundation.commuting-squares-of-maps
+open import foundation.contractible-types
+open import foundation.dependent-pair-types
+open import foundation.equality-dependent-function-types
+open import foundation.equivalences
+open import foundation.function-types
+open import foundation.functoriality-dependent-pair-types
+open import foundation.fundamental-theorem-of-identity-types
+open import foundation.homotopies
+open import foundation.homotopy-induction
+open import foundation.identity-types
+open import foundation.implicit-function-types
+open import foundation.morphisms-arrows
+open import foundation.precomposition-functions
+open import foundation.propositions
+open import foundation.pullbacks
+open import foundation.sections
+open import foundation.structure-identity-principle
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+open import foundation.whiskering-homotopies-composition
+
+open import foundation-core.equality-dependent-pair-types
+open import foundation-core.retractions
+open import foundation-core.torsorial-type-families
+
+open import trees.natural-transformations-polynomial-endofunctors
+open import trees.polynomial-endofunctors
+```
+
+</details>
+
+## Idea
+
+Given two [polynomial endofunctors](trees.polynomial-endofunctors.md)
+$𝑃 ≐ (A ◃ B)$ and $𝑄 ≐ (C ◃ D)$, then a
+[natural transformation](trees.natural-transformations-polynomial-endofunctors.md)
+$α$ between them is
+{{#concept "cartesian" Disambiguation="natural transformations of polynomial endofunctors of types" Agda=is-cartesian-natural-transformation-polynomial-endofunctor}}
+
+if every naturality square
+
+```text
+              α(X)
+     𝑃(X) -----------> 𝑄(X)
+       |                |
+       |                |
+  𝑃(f) |                | 𝑄(f)
+       |                |
+       ∨                ∨
+     𝑃(Y) -----------> 𝑄(Y)
+              α(Y)
+```
+
+is a [pullback](foundation-core.pullbacks.md).
+
+## Definitions
+
+### The cartesianness predicate on natural transformations between polynomial endofunctors
+
+```agda
+module _
+  {l1 l2 l3 l4 l : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : natural-transformation-polynomial-endofunctor l 𝑃 𝑄)
+  where
+
+  is-cartesian-natural-transformation-polynomial-endofunctor :
+    UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ lsuc l)
+  is-cartesian-natural-transformation-polynomial-endofunctor =
+    {X Y : UU l} (f : X → Y) →
+    is-cartesian-hom-arrow
+      ( map-polynomial-endofunctor 𝑃 f)
+      ( map-polynomial-endofunctor 𝑄 f)
+      ( hom-arrow-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α f)
+
+  is-prop-is-cartesian-natural-transformation-polynomial-endofunctor :
+    is-prop is-cartesian-natural-transformation-polynomial-endofunctor
+  is-prop-is-cartesian-natural-transformation-polynomial-endofunctor =
+    is-prop-implicit-Π
+      ( λ X →
+        is-prop-implicit-Π
+          ( λ Y →
+            is-prop-Π
+              ( λ f →
+                is-prop-is-cartesian-hom-arrow
+                  ( map-polynomial-endofunctor 𝑃 f)
+                  ( map-polynomial-endofunctor 𝑄 f)
+                  ( hom-arrow-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+                    ( α)
+                    ( f)))))
+
+  is-cartesian-natural-transformation-polynomial-endofunctor-Prop :
+    Prop (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ lsuc l)
+  is-cartesian-natural-transformation-polynomial-endofunctor-Prop =
+    ( is-cartesian-natural-transformation-polynomial-endofunctor ,
+      is-prop-is-cartesian-natural-transformation-polynomial-endofunctor)
+```
+
+### Cartesian natural transformations between polynomial endofunctors
+
+```agda
+cartesian-natural-transformation-polynomial-endofunctor :
+  {l1 l2 l3 l4 : Level} (l : Level) →
+  polynomial-endofunctor l1 l2 →
+  polynomial-endofunctor l3 l4 →
+  UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ lsuc l)
+cartesian-natural-transformation-polynomial-endofunctor l 𝑃 𝑄 =
+  Σ ( natural-transformation-polynomial-endofunctor l 𝑃 𝑄)
+    ( is-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄)
+
+module _
+  {l1 l2 l3 l4 l : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : cartesian-natural-transformation-polynomial-endofunctor l 𝑃 𝑄)
+  where
+
+  natural-transformation-cartesian-natural-transformation-polynomial-endofunctor :
+    natural-transformation-polynomial-endofunctor l 𝑃 𝑄
+  natural-transformation-cartesian-natural-transformation-polynomial-endofunctor =
+    pr1 α
+
+  type-cartesian-natural-transformation-polynomial-endofunctor :
+    {X : UU l} →
+    type-polynomial-endofunctor 𝑃 X →
+    type-polynomial-endofunctor 𝑄 X
+  type-cartesian-natural-transformation-polynomial-endofunctor =
+    type-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
+
+  naturality-cartesian-natural-transformation-polynomial-endofunctor :
+    coherence-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( type-cartesian-natural-transformation-polynomial-endofunctor)
+  naturality-cartesian-natural-transformation-polynomial-endofunctor =
+    naturality-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
+
+  is-cartesian-cartesian-natural-transformation-polynomial-endofunctor :
+    is-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
+  is-cartesian-cartesian-natural-transformation-polynomial-endofunctor = pr2 α
+```
+
+### The associated family of cartesian morphisms of arrows
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α@(α' , H) : cartesian-natural-transformation-polynomial-endofunctor l5 𝑃 𝑄)
+  where
+
+  hom-arrow-cartesian-natural-transformation-polynomial-endofunctor :
+    {X Y : UU l5} (f : X → Y) →
+    hom-arrow (map-polynomial-endofunctor 𝑃 f) (map-polynomial-endofunctor 𝑄 f)
+  hom-arrow-cartesian-natural-transformation-polynomial-endofunctor =
+    hom-arrow-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α'
+
+  cartesian-hom-arrow-cartesian-natural-transformation-polynomial-endofunctor :
+    {X Y : UU l5} (f : X → Y) →
+    cartesian-hom-arrow
+      ( map-polynomial-endofunctor 𝑃 f)
+      ( map-polynomial-endofunctor 𝑄 f)
+  cartesian-hom-arrow-cartesian-natural-transformation-polynomial-endofunctor
+    f =
+    ( hom-arrow-cartesian-natural-transformation-polynomial-endofunctor f , H f)
+```
+
+### The identity cartesian natural transformation
+
+```agda
+id-cartesian-natural-transformation-polynomial-endofunctor :
+  {l1 l2 l3 : Level} (𝑃 : polynomial-endofunctor l1 l2) →
+  cartesian-natural-transformation-polynomial-endofunctor l3 𝑃 𝑃
+pr1 (id-cartesian-natural-transformation-polynomial-endofunctor 𝑃) =
+  id-natural-transformation-polynomial-endofunctor 𝑃
+pr2 (id-cartesian-natural-transformation-polynomial-endofunctor 𝑃) f =
+  is-cartesian-id-hom-arrow
+```
+
+### Composition of cartesian natural transformations
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 l6 l : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (𝑅 : polynomial-endofunctor l5 l6)
+  (β@(β' , H) : cartesian-natural-transformation-polynomial-endofunctor l 𝑄 𝑅)
+  (α@(α' , K) : cartesian-natural-transformation-polynomial-endofunctor l 𝑃 𝑄)
+  where
+
+  is-cartesian-comp-cartesian-natural-transformation-polynomial-endofunctor :
+    is-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑅
+      ( comp-natural-transformation-polynomial-endofunctor 𝑃 𝑄 𝑅 β' α')
+  is-cartesian-comp-cartesian-natural-transformation-polynomial-endofunctor f =
+    is-cartesian-comp-hom-arrow
+      ( map-polynomial-endofunctor 𝑃 f)
+      ( map-polynomial-endofunctor 𝑄 f)
+      ( map-polynomial-endofunctor 𝑅 f)
+      ( hom-arrow-natural-transformation-polynomial-endofunctor 𝑄 𝑅 β' f)
+      ( hom-arrow-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α' f)
+      ( H f)
+      ( K f)
+
+  comp-cartesian-natural-transformation-polynomial-endofunctor :
+    cartesian-natural-transformation-polynomial-endofunctor l 𝑃 𝑅
+  comp-cartesian-natural-transformation-polynomial-endofunctor =
+    ( comp-natural-transformation-polynomial-endofunctor 𝑃 𝑄 𝑅 β' α' ,
+      is-cartesian-comp-cartesian-natural-transformation-polynomial-endofunctor)
+```

From 5ec4c5878df64f29d892fea6f9fb51229c1e40ec Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sun, 31 Aug 2025 22:18:04 +0200
Subject: [PATCH 06/41] work

---
 ...morphisms-polynomial-endofunctors.lagda.md | 28 +++++++++++++++++--
 src/trees/polynomial-endofunctors.lagda.md    |  2 ++
 2 files changed, 28 insertions(+), 2 deletions(-)

diff --git a/src/trees/morphisms-polynomial-endofunctors.lagda.md b/src/trees/morphisms-polynomial-endofunctors.lagda.md
index e2f0208da4..5770feaa87 100644
--- a/src/trees/morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/morphisms-polynomial-endofunctors.lagda.md
@@ -89,6 +89,31 @@ module _
       ( λ a → precomp (positions-hom-polynomial-endofunctor a) X)
 ```
 
+### The identity morphism
+
+```agda
+id-hom-polynomial-endofunctor :
+  {l1 l2 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2) →
+  hom-polynomial-endofunctor 𝑃 𝑃
+id-hom-polynomial-endofunctor 𝑃 = (id , (λ a → id))
+```
+
+### Composition of morphisms
+
+```agda
+comp-hom-polynomial-endofunctor :
+  {l1 l2 l3 l4 l5 l6 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (𝑅 : polynomial-endofunctor l5 l6) →
+  hom-polynomial-endofunctor 𝑄 𝑅 →
+  hom-polynomial-endofunctor 𝑃 𝑄 →
+  hom-polynomial-endofunctor 𝑃 𝑅
+comp-hom-polynomial-endofunctor 𝑃 𝑄 𝑅 (β₀ , β₁) (α₀ , α₁) =
+  ( β₀ ∘ α₀ , (λ a → α₁ a ∘ β₁ (α₀ a)))
+```
+
 ## Properties
 
 ### Characterizing equality of morphisms
@@ -217,8 +242,7 @@ module _
   where
 
   shapes-natural-transformation-polynomial-endofunctor :
-    shapes-polynomial-endofunctor 𝑃 →
-    shapes-polynomial-endofunctor 𝑄
+    shapes-polynomial-endofunctor 𝑃 → shapes-polynomial-endofunctor 𝑄
   shapes-natural-transformation-polynomial-endofunctor a =
     pr1 (type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α (a , id))
 
diff --git a/src/trees/polynomial-endofunctors.lagda.md b/src/trees/polynomial-endofunctors.lagda.md
index 5479e449a0..52534d3473 100644
--- a/src/trees/polynomial-endofunctors.lagda.md
+++ b/src/trees/polynomial-endofunctors.lagda.md
@@ -204,3 +204,5 @@ coh-refl-htpy-polynomial-endofunctor (A , B) =
   In other words, given a [species of types](species.species-of-types.md) `S`,
   the shapes are types equipped with `S`-structure, and the positions are
   points.
+- Via [type duality](foundation.type-duality.md), polynomial endofunctors are
+  classified by maps of types.

From 4683b4b0e3b26ada0cbf19a548d1d2e2b6f60732 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Tue, 2 Sep 2025 00:19:17 +0200
Subject: [PATCH 07/41] work cart poly fun

---
 src/trees.lagda.md                            |   2 +
 ...morphisms-polynomial-endofunctors.lagda.md | 300 ++++++++++++++++++
 src/trees/polynomial-endofunctors.lagda.md    |   2 +-
 3 files changed, 303 insertions(+), 1 deletion(-)
 create mode 100644 src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md

diff --git a/src/trees.lagda.md b/src/trees.lagda.md
index a3fad6bbf4..be9ef14d32 100644
--- a/src/trees.lagda.md
+++ b/src/trees.lagda.md
@@ -14,6 +14,7 @@ open import trees.bases-directed-trees public
 open import trees.bases-enriched-directed-trees public
 open import trees.binary-w-types public
 open import trees.bounded-multisets public
+open import trees.cartesian-morphisms-polynomial-endofunctors public
 open import trees.cartesian-natural-transformations-polynomial-endofunctors public
 open import trees.coalgebra-of-directed-trees public
 open import trees.coalgebra-of-enriched-directed-trees public
@@ -51,6 +52,7 @@ open import trees.natural-transformations-polynomial-endofunctors public
 open import trees.planar-binary-trees public
 open import trees.plane-trees public
 open import trees.polynomial-endofunctors public
+open import trees.polynomial-endofunctors-from-subuniverses public
 open import trees.raising-universe-levels-directed-trees public
 open import trees.ranks-of-elements-w-types public
 open import trees.rooted-morphisms-directed-trees public
diff --git a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
new file mode 100644
index 0000000000..47a81df38f
--- /dev/null
+++ b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
@@ -0,0 +1,300 @@
+# Cartesian morphisms of polynomial endofunctors
+
+```agda
+module trees.cartesian-morphisms-polynomial-endofunctors where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.action-on-identifications-functions
+open import foundation.binary-homotopies
+open import foundation.commuting-squares-of-homotopies
+open import foundation.commuting-squares-of-maps
+open import foundation.contractible-types
+open import foundation.dependent-pair-types
+open import foundation.equality-dependent-function-types
+open import foundation.equivalences
+open import foundation.function-types
+open import foundation.functoriality-dependent-pair-types
+open import foundation.fundamental-theorem-of-identity-types
+open import foundation.homotopies
+open import foundation.homotopy-induction
+open import foundation.identity-types
+open import foundation.implicit-function-types
+open import foundation.precomposition-functions
+open import foundation.propositions
+open import foundation.sections
+open import foundation.structure-identity-principle
+open import foundation.subtypes
+open import foundation.transport-along-identifications
+open import foundation.truncated-types
+open import foundation.truncation-levels
+open import foundation.type-arithmetic-dependent-pair-types
+open import foundation.type-theoretic-principle-of-choice
+open import foundation.universe-levels
+open import foundation.whiskering-homotopies-composition
+
+open import foundation-core.equality-dependent-pair-types
+open import foundation-core.retractions
+open import foundation-core.torsorial-type-families
+
+open import trees.cartesian-natural-transformations-polynomial-endofunctors
+open import trees.morphisms-polynomial-endofunctors
+open import trees.natural-transformations-polynomial-endofunctors
+open import trees.polynomial-endofunctors
+```
+
+</details>
+
+## Idea
+
+Given two [polynomial endofunctors](trees.polynomial-endofunctors.md)
+$𝑃 ≐ (A ◃ B)$ and $𝑄 ≐ (C ◃ D)$, a
+[morphism](trees.morphisms-polynomial-endofunctors.md) $α : 𝑃 → 𝑄$ is
+{{#concept "cartesian" Disambiguation="morphism of polynomial endofunctors of types" Agda=is-cartesian-hom-polynomial-endofunctor Agda=cartesian-hom-polynomial-endofunctor}}
+if the family of maps
+
+$$α₁ : (a : A) → D (α₀ a) → B a$$
+
+is a family of [equivalences](foundation-core.equivalences.md).
+
+## Definitions
+
+### The predicate of being cartesian
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : hom-polynomial-endofunctor 𝑃 𝑄)
+  where
+
+  is-cartesian-hom-polynomial-endofunctor : UU (l1 ⊔ l2 ⊔ l4)
+  is-cartesian-hom-polynomial-endofunctor =
+    (a : shapes-polynomial-endofunctor 𝑃) →
+    is-equiv (positions-hom-polynomial-endofunctor 𝑃 𝑄 α a)
+
+  is-prop-is-cartesian-hom-polynomial-endofunctor :
+    is-prop is-cartesian-hom-polynomial-endofunctor
+  is-prop-is-cartesian-hom-polynomial-endofunctor =
+    is-prop-Π
+      ( λ a →
+        is-property-is-equiv (positions-hom-polynomial-endofunctor 𝑃 𝑄 α a))
+
+  is-cartesian-hom-polynomial-endofunctor-Prop : Prop (l1 ⊔ l2 ⊔ l4)
+  is-cartesian-hom-polynomial-endofunctor-Prop =
+    ( is-cartesian-hom-polynomial-endofunctor ,
+      is-prop-is-cartesian-hom-polynomial-endofunctor)
+```
+
+### The type of cartesian morphisms
+
+```agda
+cartesian-hom-polynomial-endofunctor :
+  {l1 l2 l3 l4 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4) →
+  UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
+cartesian-hom-polynomial-endofunctor 𝑃 𝑄 =
+  Σ ( hom-polynomial-endofunctor 𝑃 𝑄)
+    ( is-cartesian-hom-polynomial-endofunctor 𝑃 𝑄)
+
+make-cartesian-hom-polynomial-endofunctor :
+  {l1 l2 l3 l4 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α₀ : shapes-polynomial-endofunctor 𝑃 → shapes-polynomial-endofunctor 𝑄) →
+  ( (a : shapes-polynomial-endofunctor 𝑃) →
+    positions-polynomial-endofunctor 𝑄 (α₀ a) ≃
+    positions-polynomial-endofunctor 𝑃 a) →
+  cartesian-hom-polynomial-endofunctor 𝑃 𝑄
+make-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α₀ α₁ =
+  ( ( α₀ , map-equiv ∘ α₁) , is-equiv-map-equiv ∘ α₁)
+
+module _
+  {l1 l2 l3 l4 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : cartesian-hom-polynomial-endofunctor 𝑃 𝑄)
+  where
+
+  hom-cartesian-hom-polynomial-endofunctor : hom-polynomial-endofunctor 𝑃 𝑄
+  hom-cartesian-hom-polynomial-endofunctor = pr1 α
+
+  shapes-cartesian-hom-polynomial-endofunctor :
+    shapes-polynomial-endofunctor 𝑃 → shapes-polynomial-endofunctor 𝑄
+  shapes-cartesian-hom-polynomial-endofunctor =
+    shapes-hom-polynomial-endofunctor 𝑃 𝑄
+      hom-cartesian-hom-polynomial-endofunctor
+
+  positions-cartesian-hom-polynomial-endofunctor :
+    (a : shapes-polynomial-endofunctor 𝑃) →
+    positions-polynomial-endofunctor 𝑄
+      ( shapes-hom-polynomial-endofunctor 𝑃 𝑄
+        ( hom-cartesian-hom-polynomial-endofunctor)
+        ( a)) →
+    positions-polynomial-endofunctor 𝑃 a
+  positions-cartesian-hom-polynomial-endofunctor =
+    positions-hom-polynomial-endofunctor 𝑃 𝑄
+      hom-cartesian-hom-polynomial-endofunctor
+
+  type-cartesian-hom-polynomial-endofunctor :
+    {l5 : Level} {X : UU l5} →
+    type-polynomial-endofunctor 𝑃 X → type-polynomial-endofunctor 𝑄 X
+  type-cartesian-hom-polynomial-endofunctor =
+    type-hom-polynomial-endofunctor 𝑃 𝑄
+      hom-cartesian-hom-polynomial-endofunctor
+
+  is-cartesian-cartesian-hom-polynomial-endofunctor :
+    is-cartesian-hom-polynomial-endofunctor 𝑃 𝑄
+      hom-cartesian-hom-polynomial-endofunctor
+  is-cartesian-cartesian-hom-polynomial-endofunctor = pr2 α
+
+  equiv-positions-cartesian-hom-polynomial-endofunctor :
+    (a : shapes-polynomial-endofunctor 𝑃) →
+    positions-polynomial-endofunctor 𝑄
+      ( shapes-hom-polynomial-endofunctor 𝑃 𝑄
+        ( hom-cartesian-hom-polynomial-endofunctor)
+        ( a)) ≃
+    positions-polynomial-endofunctor 𝑃 a
+  equiv-positions-cartesian-hom-polynomial-endofunctor a =
+    ( positions-cartesian-hom-polynomial-endofunctor a ,
+      is-cartesian-cartesian-hom-polynomial-endofunctor a)
+```
+
+### The identity cartesian morphism
+
+```agda
+id-cartesian-hom-polynomial-endofunctor :
+  {l1 l2 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2) →
+  cartesian-hom-polynomial-endofunctor 𝑃 𝑃
+id-cartesian-hom-polynomial-endofunctor 𝑃 =
+   (id-hom-polynomial-endofunctor 𝑃 , (λ a → is-equiv-id))
+```
+
+### Composition of cartesian morphisms
+
+```agda
+comp-cartesian-hom-polynomial-endofunctor :
+  {l1 l2 l3 l4 l5 l6 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (𝑅 : polynomial-endofunctor l5 l6) →
+  cartesian-hom-polynomial-endofunctor 𝑄 𝑅 →
+  cartesian-hom-polynomial-endofunctor 𝑃 𝑄 →
+  cartesian-hom-polynomial-endofunctor 𝑃 𝑅
+comp-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 𝑅 (β , H) (α , K) =
+  ( comp-hom-polynomial-endofunctor 𝑃 𝑄 𝑅 β α ,
+    λ a → is-equiv-comp (pr2 α a) (pr2 β (pr1 α a)) (H (pr1 α a)) (K a))
+```
+
+## Properties
+
+### Truncatedness of the type of morphisms
+
+If the shapes and positions of the codomain $𝑄$ are $k$-truncated, for $k ≥ -1$,
+then the type of cartesian morphisms $𝑃 → 𝑄$ is $k$-truncated.
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  where
+
+  is-trunc-succ-cartesian-hom-polynomial-endofunctor :
+    (k : 𝕋) →
+    is-trunc (succ-𝕋 k) (shapes-polynomial-endofunctor 𝑄) →
+    ( (c : shapes-polynomial-endofunctor 𝑄) →
+      is-trunc (succ-𝕋 k) (positions-polynomial-endofunctor 𝑄 c)) →
+    is-trunc (succ-𝕋 k) (cartesian-hom-polynomial-endofunctor 𝑃 𝑄)
+  is-trunc-succ-cartesian-hom-polynomial-endofunctor k hQ hP =
+    is-trunc-equiv (succ-𝕋 k) _
+      ( equiv-tot (λ _ → inv-distributive-Π-Σ) ∘e associative-Σ _ _ _)
+      ( is-trunc-Σ
+        ( is-trunc-function-type (succ-𝕋 k) hQ)
+        ( λ f →
+          is-trunc-Π
+            ( succ-𝕋 k)
+            ( λ e → is-trunc-equiv-is-trunc-domain k (hP (f e)))))
+```
+
+### Cartesian morphisms are cartesian natural transformations
+
+```text
+module _
+  {l1 l2 l3 l4 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : hom-polynomial-endofunctor 𝑃 𝑄)
+  where
+
+  is-cartesian-natural-transformation-hom-polynomial-endofunctor :
+    {l : Level} →
+    is-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( natural-transformation-hom-polynomial-endofunctor 𝑃 𝑄 α {l})
+  is-cartesian-natural-transformation-hom-polynomial-endofunctor f =
+    {!   !}
+```
+
+### Cartesian natural transformations define cartesian morphisms
+
+```text
+module _
+  {l1 l2 l3 l4 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : natural-transformation-polynomial-endofunctor l2 𝑃 𝑄)
+  where
+
+  shapes-natural-transformation-polynomial-endofunctor :
+    shapes-polynomial-endofunctor 𝑃 → shapes-polynomial-endofunctor 𝑄
+  shapes-natural-transformation-polynomial-endofunctor a =
+    pr1 (type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α (a , id))
+
+  positions-natural-transformation-polynomial-endofunctor :
+    (a : shapes-polynomial-endofunctor 𝑃) →
+    positions-polynomial-endofunctor 𝑄
+      ( shapes-natural-transformation-polynomial-endofunctor a) →
+    positions-polynomial-endofunctor 𝑃 a
+  positions-natural-transformation-polynomial-endofunctor a =
+    pr2 (type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α (a , id))
+
+  hom-natural-transformation-polynomial-endofunctor :
+    hom-polynomial-endofunctor 𝑃 𝑄
+  hom-natural-transformation-polynomial-endofunctor =
+    ( shapes-natural-transformation-polynomial-endofunctor ,
+      positions-natural-transformation-polynomial-endofunctor)
+```
+
+### Equivalence between cartesian morphisms and cartesian natural transformations
+
+```text
+module _
+  {l1 l2 l3 l4 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  where
+
+  is-retraction-hom-natural-transformation-polynomial-endofunctor :
+    is-retraction
+      ( λ α → natural-transformation-hom-polynomial-endofunctor 𝑃 𝑄 α {l2})
+      ( hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄)
+  is-retraction-hom-natural-transformation-polynomial-endofunctor α = refl
+
+  is-section-type-hom-natural-transformation-polynomial-endofunctor :
+    (α : natural-transformation-polynomial-endofunctor l2 𝑃 𝑄)
+    (X : UU l2) →
+    type-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( natural-transformation-hom-polynomial-endofunctor 𝑃 𝑄
+        ( hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α)) ~
+    type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α {X}
+  is-section-type-hom-natural-transformation-polynomial-endofunctor
+    α X (a , x) =
+    inv
+      ( naturality-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α x
+        ( a , id))
+```
diff --git a/src/trees/polynomial-endofunctors.lagda.md b/src/trees/polynomial-endofunctors.lagda.md
index 52534d3473..ebfea18c3c 100644
--- a/src/trees/polynomial-endofunctors.lagda.md
+++ b/src/trees/polynomial-endofunctors.lagda.md
@@ -35,7 +35,7 @@ Given a type `A` [equipped](foundation.structure.md) with a type family `B` over
 `𝑃 A B` is defined by
 
 ```text
-  X ↦ Σ (x : A), (B x → X)
+  X ↦ Σ (x : A), (B x → X).
 ```
 
 Polynomial endofunctors are important in the study of

From 2bce3f4a119715a4e4b2fc88796d6761e4079cda Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Tue, 2 Sep 2025 00:31:45 +0200
Subject: [PATCH 08/41] edit

---
 ...morphisms-polynomial-endofunctors.lagda.md | 53 +++++++++++++++----
 1 file changed, 44 insertions(+), 9 deletions(-)

diff --git a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
index 47a81df38f..5048854bf2 100644
--- a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
@@ -193,6 +193,30 @@ comp-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 𝑅 (β , H) (α , K) =
 
 ## Properties
 
+### A computation of the type of cartesian morphisms
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  where
+
+  cartesian-hom-polynomial-endofunctor' : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
+  cartesian-hom-polynomial-endofunctor' =
+    Σ ( shapes-polynomial-endofunctor 𝑃 → shapes-polynomial-endofunctor 𝑄)
+      ( λ α₀ →
+        ((a : shapes-polynomial-endofunctor 𝑃) →
+          positions-polynomial-endofunctor 𝑄 (α₀ a) ≃
+          positions-polynomial-endofunctor 𝑃 a))
+
+  reassociate-type-cartesian-hom-polynomial-endofunctor :
+    cartesian-hom-polynomial-endofunctor 𝑃 𝑄 ≃
+    cartesian-hom-polynomial-endofunctor'
+  reassociate-type-cartesian-hom-polynomial-endofunctor =
+    ( equiv-tot (λ _ → inv-distributive-Π-Σ)) ∘e (associative-Σ _ _ _)
+```
+
 ### Truncatedness of the type of morphisms
 
 If the shapes and positions of the codomain $𝑄$ are $k$-truncated, for $k ≥ -1$,
@@ -205,21 +229,32 @@ module _
   (𝑄 : polynomial-endofunctor l3 l4)
   where
 
+  is-trunc-succ-cartesian-hom-polynomial-endofunctor' :
+    (k : 𝕋) →
+    is-trunc (succ-𝕋 k) (shapes-polynomial-endofunctor 𝑄) →
+    ( (c : shapes-polynomial-endofunctor 𝑄) →
+      is-trunc (succ-𝕋 k) (positions-polynomial-endofunctor 𝑄 c)) →
+    is-trunc (succ-𝕋 k) (cartesian-hom-polynomial-endofunctor' 𝑃 𝑄)
+  is-trunc-succ-cartesian-hom-polynomial-endofunctor' k hQ hQ' =
+    is-trunc-Σ
+      ( is-trunc-function-type (succ-𝕋 k) hQ)
+      ( λ f →
+        is-trunc-Π
+          ( succ-𝕋 k)
+          ( λ e → is-trunc-equiv-is-trunc-domain k (hQ' (f e))))
+
   is-trunc-succ-cartesian-hom-polynomial-endofunctor :
     (k : 𝕋) →
     is-trunc (succ-𝕋 k) (shapes-polynomial-endofunctor 𝑄) →
     ( (c : shapes-polynomial-endofunctor 𝑄) →
       is-trunc (succ-𝕋 k) (positions-polynomial-endofunctor 𝑄 c)) →
     is-trunc (succ-𝕋 k) (cartesian-hom-polynomial-endofunctor 𝑃 𝑄)
-  is-trunc-succ-cartesian-hom-polynomial-endofunctor k hQ hP =
-    is-trunc-equiv (succ-𝕋 k) _
-      ( equiv-tot (λ _ → inv-distributive-Π-Σ) ∘e associative-Σ _ _ _)
-      ( is-trunc-Σ
-        ( is-trunc-function-type (succ-𝕋 k) hQ)
-        ( λ f →
-          is-trunc-Π
-            ( succ-𝕋 k)
-            ( λ e → is-trunc-equiv-is-trunc-domain k (hP (f e)))))
+  is-trunc-succ-cartesian-hom-polynomial-endofunctor k hQ hQ' =
+    is-trunc-equiv
+      ( succ-𝕋 k)
+      ( cartesian-hom-polynomial-endofunctor' 𝑃 𝑄)
+      ( reassociate-type-cartesian-hom-polynomial-endofunctor 𝑃 𝑄)
+      ( is-trunc-succ-cartesian-hom-polynomial-endofunctor' k hQ hQ')
 ```
 
 ### Cartesian morphisms are cartesian natural transformations

From 3703bc145416879884af4e52a0254693f4c5dea7 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Tue, 2 Sep 2025 16:15:32 +0200
Subject: [PATCH 09/41] fix references that should be citations in
 `coherently-idempotent-maps`

---
 src/foundation/coherently-idempotent-maps.lagda.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/foundation/coherently-idempotent-maps.lagda.md b/src/foundation/coherently-idempotent-maps.lagda.md
index d746a1ec6a..c88e1bf662 100644
--- a/src/foundation/coherently-idempotent-maps.lagda.md
+++ b/src/foundation/coherently-idempotent-maps.lagda.md
@@ -46,9 +46,9 @@ given in {{#cite Shu17}}:
 ```
 
 **Terminology.** Our definition of a _coherently idempotent map_ corresponds to
-the definition of a _(fully coherent) idempotent map_ in {{#reference Shu17}}
-and {{#reference Shu14SplittingIdempotents}}. Our definition of an _idempotent
-map_ corresponds in their terminology to a _pre-idempotent map_.
+the definition of a _(fully coherent) idempotent map_ in {{#cite Shu17}} and
+{{#cite Shu14SplittingIdempotents}}. Our definition of an _idempotent map_
+corresponds in their terminology to a _pre-idempotent map_.
 
 ## Definitions
 

From 1ee4e3e4f0170db65660f9b9040793da708ac265 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 13 Sep 2025 12:30:04 +0200
Subject: [PATCH 10/41] fiber computations

---
 .../dependent-sums-pullbacks.lagda.md         |  3 +-
 src/trees.lagda.md                            |  1 -
 ...morphisms-polynomial-endofunctors.lagda.md | 92 +++++++++++++++++++
 src/trees/polynomial-endofunctors.lagda.md    | 27 ++++++
 4 files changed, 120 insertions(+), 3 deletions(-)

diff --git a/src/foundation/dependent-sums-pullbacks.lagda.md b/src/foundation/dependent-sums-pullbacks.lagda.md
index c1f09588ee..eb6bd139dc 100644
--- a/src/foundation/dependent-sums-pullbacks.lagda.md
+++ b/src/foundation/dependent-sums-pullbacks.lagda.md
@@ -283,8 +283,7 @@ module _
 
   is-pullback-family-is-pullback-tot :
     is-pullback f g c →
-    is-pullback
-      (map-Σ PX f f') (map-Σ PX g g') tot-cone-cone-family →
+    is-pullback (map-Σ PX f f') (map-Σ PX g g') tot-cone-cone-family →
     (x : C) →
     is-pullback
       ( ( tr PX (coherence-square-cone f g c x)) ∘
diff --git a/src/trees.lagda.md b/src/trees.lagda.md
index be9ef14d32..5d2a613423 100644
--- a/src/trees.lagda.md
+++ b/src/trees.lagda.md
@@ -52,7 +52,6 @@ open import trees.natural-transformations-polynomial-endofunctors public
 open import trees.planar-binary-trees public
 open import trees.plane-trees public
 open import trees.polynomial-endofunctors public
-open import trees.polynomial-endofunctors-from-subuniverses public
 open import trees.raising-universe-levels-directed-trees public
 open import trees.ranks-of-elements-w-types public
 open import trees.rooted-morphisms-directed-trees public
diff --git a/src/trees/morphisms-polynomial-endofunctors.lagda.md b/src/trees/morphisms-polynomial-endofunctors.lagda.md
index 5770feaa87..901315003f 100644
--- a/src/trees/morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/morphisms-polynomial-endofunctors.lagda.md
@@ -11,10 +11,13 @@ open import foundation.action-on-identifications-functions
 open import foundation.binary-homotopies
 open import foundation.commuting-squares-of-homotopies
 open import foundation.commuting-squares-of-maps
+open import foundation.commuting-triangles-of-maps
+open import foundation.cones-over-cospan-diagrams
 open import foundation.contractible-types
 open import foundation.dependent-pair-types
 open import foundation.equality-dependent-function-types
 open import foundation.equivalences
+open import foundation.fibers-of-maps
 open import foundation.function-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.fundamental-theorem-of-identity-types
@@ -22,6 +25,7 @@ open import foundation.homotopies
 open import foundation.homotopy-induction
 open import foundation.identity-types
 open import foundation.implicit-function-types
+open import foundation.morphisms-arrows
 open import foundation.precomposition-functions
 open import foundation.sections
 open import foundation.structure-identity-principle
@@ -229,6 +233,26 @@ module _
   natural-transformation-hom-polynomial-endofunctor =
     ( type-hom-polynomial-endofunctor 𝑃 𝑄 α ,
       naturality-hom-polynomial-endofunctor)
+
+  hom-arrow-hom-polynomial-endofunctor :
+    {l5 l6 : Level} {X : UU l5} {Y : UU l6} (f : X → Y) →
+    hom-arrow (map-polynomial-endofunctor 𝑃 f) (map-polynomial-endofunctor 𝑄 f)
+  hom-arrow-hom-polynomial-endofunctor f =
+    ( type-hom-polynomial-endofunctor 𝑃 𝑄 α ,
+      type-hom-polynomial-endofunctor 𝑃 𝑄 α ,
+      naturality-hom-polynomial-endofunctor f)
+
+  cone-hom-polynomial-endofunctor :
+    {l5 l6 : Level} {X : UU l5} {Y : UU l6} (f : X → Y) →
+    cone
+      ( type-hom-polynomial-endofunctor 𝑃 𝑄 α)
+      ( map-polynomial-endofunctor 𝑄 f)
+      ( type-polynomial-endofunctor 𝑃 X)
+  cone-hom-polynomial-endofunctor f =
+    cone-hom-arrow
+      ( map-polynomial-endofunctor 𝑃 f)
+      ( map-polynomial-endofunctor 𝑄 f)
+      ( hom-arrow-hom-polynomial-endofunctor f)
 ```
 
 ### Natural transformations define morphisms
@@ -261,6 +285,74 @@ module _
       positions-natural-transformation-polynomial-endofunctor)
 ```
 
+### Computing the fibers of the induced natural transformation
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α@(α₀ , α₁) : hom-polynomial-endofunctor 𝑃 𝑄)
+  (let 𝑃₁ = positions-polynomial-endofunctor 𝑃)
+  (let 𝑄₀ = shapes-polynomial-endofunctor 𝑄)
+  (let 𝑄₁ = positions-polynomial-endofunctor 𝑄)
+  {X : UU l5}
+  where
+
+  fiber-type-hom-polynomial-endofunctor :
+    (c : 𝑄₀) (x : 𝑄₁ c → X) → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l5)
+  fiber-type-hom-polynomial-endofunctor c x =
+    Σ ( fiber α₀ c)
+      ( λ (a , p) →
+        Σ (𝑃₁ a → X) (λ x' → coherence-square-maps (tr 𝑄₁ p) (α₁ a) x x'))
+
+  compute-fiber-type-hom-polynomial-endofunctor :
+    (c : 𝑄₀) (x : 𝑄₁ c → X) →
+    fiber (type-hom-polynomial-endofunctor 𝑃 𝑄 α {X = X}) (c , x) ≃
+    fiber-type-hom-polynomial-endofunctor c x
+  compute-fiber-type-hom-polynomial-endofunctor c x =
+    equivalence-reasoning
+      fiber (type-hom-polynomial-endofunctor 𝑃 𝑄 α {X = X}) (c , x)
+      ≃ Σ ( fiber α₀ c)
+          ( λ (a , p) →
+            fiber
+              ( precomp (α₁ a) X)
+              ( inv-tr (λ c' → pr2 𝑄 c' → X) p x))
+        by
+          compute-fiber-map-Σ
+            ( λ c → positions-polynomial-endofunctor 𝑄 c → X)
+            ( α₀)
+            ( λ a → precomp (α₁ a) X)
+            ( c , x)
+      ≃ Σ ( fiber α₀ c)
+          ( λ (a , p) →
+            Σ (𝑃₁ a → X)
+              (λ x' →
+                coherence-triangle-maps'
+                  ( inv-tr (λ c' → 𝑄₁ c' → X) p x)
+                  ( x')
+                  ( α₁ a)))
+        by
+          equiv-tot
+            ( λ (a , p) →
+              compute-coherence-triangle-fiber-precomp'
+                ( α₁ a)
+                ( X)
+                ( inv-tr (λ c' → pr2 𝑄 c' → X) p x))
+      ≃ Σ ( fiber α₀ c)
+          ( λ (a , p) →
+            Σ (𝑃₁ a → X) (λ x' → coherence-square-maps (tr 𝑄₁ p) (α₁ a) x x'))
+        by
+          equiv-tot
+            ( λ (a , p) →
+              equiv-tot
+                ( λ x' →
+                  equiv-tr
+                    ( λ u → coherence-triangle-maps' u x' (α₁ a))
+                    ( ( tr-function-type-fixed-codomain 𝑄₁ X (inv p) x) ∙
+                      ( ap (λ q → x ∘ tr 𝑄₁ q) (inv-inv p)))))
+```
+
 ### Equivalence between morphisms and natural transformations
 
 ```agda
diff --git a/src/trees/polynomial-endofunctors.lagda.md b/src/trees/polynomial-endofunctors.lagda.md
index ebfea18c3c..6139f5a1b8 100644
--- a/src/trees/polynomial-endofunctors.lagda.md
+++ b/src/trees/polynomial-endofunctors.lagda.md
@@ -10,12 +10,14 @@ module trees.polynomial-endofunctors where
 open import foundation.contractible-types
 open import foundation.dependent-pair-types
 open import foundation.equivalences
+open import foundation.fibers-of-maps
 open import foundation.function-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.homotopy-induction
 open import foundation.identity-types
+open import foundation.postcomposition-functions
 open import foundation.structure-identity-principle
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
@@ -192,6 +194,31 @@ coh-refl-htpy-polynomial-endofunctor (A , B) =
   coh-refl-htpy-polynomial-endofunctor' A B
 ```
 
+### Computing the fiber of the action on maps
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level} (𝑃 : polynomial-endofunctor l1 l2)
+  {X : UU l3} {Y : UU l4} (f : X → Y)
+  where
+
+  compute-fiber-map-polynomial-endofunctor :
+    (a : shapes-polynomial-endofunctor 𝑃)
+    (y : positions-polynomial-endofunctor 𝑃 a → Y) →
+    fiber (map-polynomial-endofunctor 𝑃 f) (a , y) ≃
+    ( (b : positions-polynomial-endofunctor 𝑃 a) → fiber f (y b))
+  compute-fiber-map-polynomial-endofunctor a y =
+    equivalence-reasoning
+    fiber (map-polynomial-endofunctor 𝑃 f) (a , y)
+    ≃ fiber (postcomp (positions-polynomial-endofunctor 𝑃 a) f) y
+      by
+        compute-fiber-tot
+          ( λ a → postcomp (positions-polynomial-endofunctor 𝑃 a) f)
+          ( a , y)
+    ≃ ((b : positions-polynomial-endofunctor 𝑃 a) → fiber f (y b))
+      by inv-compute-Π-fiber-postcomp (positions-polynomial-endofunctor 𝑃 a) f y
+```
+
 ## See also
 
 - [Multivariable polynomial functors](trees.multivariable-polynomial-functors.md)

From 330bf02a6e15f2757f9c2267dcb14f329e1d0fa1 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 13 Sep 2025 12:54:04 +0200
Subject: [PATCH 11/41] Cartesian morphisms are cartesian natural
 transformations

---
 ...morphisms-polynomial-endofunctors.lagda.md | 123 ++++++++++++++++--
 ...ormations-polynomial-endofunctors.lagda.md |  25 +++-
 2 files changed, 132 insertions(+), 16 deletions(-)

diff --git a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
index 5048854bf2..7df96d5e4d 100644
--- a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
@@ -11,19 +11,26 @@ open import foundation.action-on-identifications-functions
 open import foundation.binary-homotopies
 open import foundation.commuting-squares-of-homotopies
 open import foundation.commuting-squares-of-maps
+open import foundation.cones-over-cospan-diagrams
+open import foundation.contractible-maps
 open import foundation.contractible-types
 open import foundation.dependent-pair-types
 open import foundation.equality-dependent-function-types
 open import foundation.equivalences
+open import foundation.equivalences-arrows
+open import foundation.fibers-of-maps
 open import foundation.function-types
 open import foundation.functoriality-dependent-pair-types
+open import foundation.functoriality-fibers-of-maps
 open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.homotopy-induction
 open import foundation.identity-types
 open import foundation.implicit-function-types
+open import foundation.morphisms-arrows
 open import foundation.precomposition-functions
 open import foundation.propositions
+open import foundation.pullbacks
 open import foundation.sections
 open import foundation.structure-identity-principle
 open import foundation.subtypes
@@ -32,6 +39,7 @@ open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-theoretic-principle-of-choice
+open import foundation.universal-property-equivalences
 open import foundation.universe-levels
 open import foundation.whiskering-homotopies-composition
 
@@ -162,6 +170,39 @@ module _
   equiv-positions-cartesian-hom-polynomial-endofunctor a =
     ( positions-cartesian-hom-polynomial-endofunctor a ,
       is-cartesian-cartesian-hom-polynomial-endofunctor a)
+
+  naturality-cartesian-hom-polynomial-endofunctor :
+    {l5 l6 : Level} {X : UU l5} {Y : UU l6} (f : X → Y) →
+    coherence-square-maps
+      ( map-polynomial-endofunctor 𝑃 f)
+      ( type-cartesian-hom-polynomial-endofunctor)
+      ( type-cartesian-hom-polynomial-endofunctor)
+      ( map-polynomial-endofunctor 𝑄 f)
+  naturality-cartesian-hom-polynomial-endofunctor =
+    naturality-hom-polynomial-endofunctor 𝑃 𝑄
+      hom-cartesian-hom-polynomial-endofunctor
+
+  natural-transformation-cartesian-hom-polynomial-endofunctor :
+    {l : Level} → natural-transformation-polynomial-endofunctor l 𝑃 𝑄
+  natural-transformation-cartesian-hom-polynomial-endofunctor =
+    natural-transformation-hom-polynomial-endofunctor 𝑃 𝑄
+      hom-cartesian-hom-polynomial-endofunctor
+
+  hom-arrow-cartesian-hom-polynomial-endofunctor :
+    {l5 l6 : Level} {X : UU l5} {Y : UU l6} (f : X → Y) →
+    hom-arrow (map-polynomial-endofunctor 𝑃 f) (map-polynomial-endofunctor 𝑄 f)
+  hom-arrow-cartesian-hom-polynomial-endofunctor =
+    hom-arrow-hom-polynomial-endofunctor 𝑃 𝑄
+      hom-cartesian-hom-polynomial-endofunctor
+
+  cone-cartesian-hom-polynomial-endofunctor :
+    {l5 l6 : Level} {X : UU l5} {Y : UU l6} (f : X → Y) →
+    cone
+      ( type-cartesian-hom-polynomial-endofunctor)
+      ( map-polynomial-endofunctor 𝑄 f)
+      ( type-polynomial-endofunctor 𝑃 X)
+  cone-cartesian-hom-polynomial-endofunctor =
+    cone-hom-polynomial-endofunctor 𝑃 𝑄 hom-cartesian-hom-polynomial-endofunctor
 ```
 
 ### The identity cartesian morphism
@@ -172,7 +213,7 @@ id-cartesian-hom-polynomial-endofunctor :
   (𝑃 : polynomial-endofunctor l1 l2) →
   cartesian-hom-polynomial-endofunctor 𝑃 𝑃
 id-cartesian-hom-polynomial-endofunctor 𝑃 =
-   (id-hom-polynomial-endofunctor 𝑃 , (λ a → is-equiv-id))
+  ( id-hom-polynomial-endofunctor 𝑃 , (λ a → is-equiv-id))
 ```
 
 ### Composition of cartesian morphisms
@@ -217,7 +258,7 @@ module _
     ( equiv-tot (λ _ → inv-distributive-Π-Σ)) ∘e (associative-Σ _ _ _)
 ```
 
-### Truncatedness of the type of morphisms
+### Truncatedness of the type of cartesian morphisms
 
 If the shapes and positions of the codomain $𝑄$ are $k$-truncated, for $k ≥ -1$,
 then the type of cartesian morphisms $𝑃 → 𝑄$ is $k$-truncated.
@@ -257,22 +298,84 @@ module _
       ( is-trunc-succ-cartesian-hom-polynomial-endofunctor' k hQ hQ')
 ```
 
+### Computing the fibers of the induced natural transformation
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : cartesian-hom-polynomial-endofunctor 𝑃 𝑄)
+  (let α₀ = shapes-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α)
+  (let α₁ = positions-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α)
+  {X : UU l5}
+  where
+
+  compute-fiber-type-cartesian-hom-polynomial-endofunctor :
+    (c : shapes-polynomial-endofunctor 𝑄)
+    (x : positions-polynomial-endofunctor 𝑄 c → X) →
+    fiber (type-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α {X = X}) (c , x) ≃
+    fiber α₀ c
+  compute-fiber-type-cartesian-hom-polynomial-endofunctor c x =
+    equivalence-reasoning
+      fiber (type-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α {X = X}) (c , x)
+      ≃ Σ ( fiber α₀ c)
+          ( λ (a , p) →
+            fiber
+              ( precomp (α₁ a) X)
+              ( inv-tr (λ c → positions-polynomial-endofunctor 𝑄 c → X) p x))
+        by
+          compute-fiber-map-Σ
+            ( λ c → positions-polynomial-endofunctor 𝑄 c → X)
+            ( α₀)
+            ( λ a → precomp (α₁ a) X)
+            ( c , x)
+      ≃ fiber α₀ c
+        by
+          right-unit-law-Σ-is-contr
+            ( λ (a , p) →
+              is-contr-map-is-equiv
+                ( is-equiv-precomp-is-equiv
+                  ( α₁ a)
+                  ( is-cartesian-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α a)
+                  ( X))
+                ( inv-tr (λ c → positions-polynomial-endofunctor 𝑄 c → X) p x))
+```
+
 ### Cartesian morphisms are cartesian natural transformations
 
-```text
+```agda
 module _
   {l1 l2 l3 l4 : Level}
   (𝑃 : polynomial-endofunctor l1 l2)
   (𝑄 : polynomial-endofunctor l3 l4)
-  (α : hom-polynomial-endofunctor 𝑃 𝑄)
+  (α : cartesian-hom-polynomial-endofunctor 𝑃 𝑄)
   where
 
-  is-cartesian-natural-transformation-hom-polynomial-endofunctor :
-    {l : Level} →
-    is-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄
-      ( natural-transformation-hom-polynomial-endofunctor 𝑃 𝑄 α {l})
-  is-cartesian-natural-transformation-hom-polynomial-endofunctor f =
-    {!   !}
+  abstract
+    is-cartesian-natural-transformation-cartesian-hom-polynomial-endofunctor :
+      {l : Level} →
+      is-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+        ( natural-transformation-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α {l})
+    is-cartesian-natural-transformation-cartesian-hom-polynomial-endofunctor f =
+      is-pullback-is-fiberwise-equiv-map-fiber-horizontal-map-cone
+        ( type-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α)
+        ( map-polynomial-endofunctor 𝑄 f)
+        ( cone-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α f)
+        ( λ (a , y) →
+          is-equiv-source-is-equiv-target-equiv-arrow
+            ( map-fiber-horizontal-map-cone
+              ( type-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α)
+              ( map-polynomial-endofunctor 𝑄 f)
+              ( cone-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α f)
+              ( a , y))
+            ( id)
+            ( ( compute-fiber-type-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α a
+                ( y)) ,
+              ( compute-fiber-type-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α a
+                ( f ∘ y)) ,
+              ( λ where (u , refl) → refl))
+            ( is-equiv-id))
 ```
 
 ### Cartesian natural transformations define cartesian morphisms
diff --git a/src/trees/natural-transformations-polynomial-endofunctors.lagda.md b/src/trees/natural-transformations-polynomial-endofunctors.lagda.md
index f1946f2c0e..72b9e168f2 100644
--- a/src/trees/natural-transformations-polynomial-endofunctors.lagda.md
+++ b/src/trees/natural-transformations-polynomial-endofunctors.lagda.md
@@ -12,6 +12,7 @@ open import foundation.action-on-identifications-functions
 open import foundation.binary-homotopies
 open import foundation.commuting-squares-of-homotopies
 open import foundation.commuting-squares-of-maps
+open import foundation.cones-over-cospan-diagrams
 open import foundation.contractible-types
 open import foundation.dependent-pair-types
 open import foundation.equality-dependent-function-types
@@ -26,6 +27,7 @@ open import foundation.identity-types
 open import foundation.implicit-function-types
 open import foundation.morphisms-arrows
 open import foundation.precomposition-functions
+open import foundation.pullbacks
 open import foundation.sections
 open import foundation.structure-identity-principle
 open import foundation.transport-along-identifications
@@ -115,15 +117,26 @@ module _
 ### The associated family of morphisms of arrows
 
 ```agda
-hom-arrow-natural-transformation-polynomial-endofunctor :
+module _
   {l1 l2 l3 l4 l5 : Level}
   (𝑃 : polynomial-endofunctor l1 l2)
   (𝑄 : polynomial-endofunctor l3 l4)
-  (α : natural-transformation-polynomial-endofunctor l5 𝑃 𝑄)
-  {X Y : UU l5} (f : X → Y) →
-  hom-arrow (map-polynomial-endofunctor 𝑃 f) (map-polynomial-endofunctor 𝑄 f)
-hom-arrow-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α@(α₀ , α₁) f =
-  ( α₀ , α₀ , α₁ f)
+  (α@(α₀ , α₁) : natural-transformation-polynomial-endofunctor l5 𝑃 𝑄)
+  {X Y : UU l5} (f : X → Y)
+  where
+
+  hom-arrow-natural-transformation-polynomial-endofunctor :
+    hom-arrow (map-polynomial-endofunctor 𝑃 f) (map-polynomial-endofunctor 𝑄 f)
+  hom-arrow-natural-transformation-polynomial-endofunctor =
+    ( α₀ , α₀ , α₁ f)
+
+  cone-natural-transformation-polynomial-endofunctor :
+    cone α₀ (map-polynomial-endofunctor 𝑄 f) (type-polynomial-endofunctor 𝑃 X)
+  cone-natural-transformation-polynomial-endofunctor =
+    cone-hom-arrow
+      ( map-polynomial-endofunctor 𝑃 f)
+      ( map-polynomial-endofunctor 𝑄 f)
+      ( hom-arrow-natural-transformation-polynomial-endofunctor)
 ```
 
 ### The identity natural transformation

From b159965d7469fd48e3a5ec71fa6d3c85644e6855 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 13 Sep 2025 14:30:38 +0200
Subject: [PATCH 12/41] the underlying global subuniverse of a univalent type
 family

---
 src/foundation/monomorphisms.lagda.md         | 16 ++++++-
 src/foundation/subuniverses.lagda.md          | 24 +++++-----
 .../univalent-type-families.lagda.md          | 45 ++++++++++++++++++-
 3 files changed, 72 insertions(+), 13 deletions(-)

diff --git a/src/foundation/monomorphisms.lagda.md b/src/foundation/monomorphisms.lagda.md
index 1908560489..278c91c435 100644
--- a/src/foundation/monomorphisms.lagda.md
+++ b/src/foundation/monomorphisms.lagda.md
@@ -65,7 +65,7 @@ module _
   (p : is-mono l3 f) {X : UU l3} (g h : X → A)
   where
 
-  equiv-postcomp-is-mono : (g ＝ h) ≃ ((f ∘ g) ＝ (f ∘ h))
+  equiv-postcomp-is-mono : (g ＝ h) ≃ (f ∘ g ＝ f ∘ h)
   pr1 equiv-postcomp-is-mono = ap (f ∘_)
   pr2 equiv-postcomp-is-mono = p X g h
 
@@ -92,6 +92,18 @@ module _
     is-emb-is-prop-map
       ( is-trunc-map-is-trunc-map-postcomp neg-one-𝕋 f
         ( λ X → is-prop-map-is-emb (is-mono-f X)))
+
+  equiv-postcomp-is-emb :
+    is-emb f →
+    {l3 : Level} {X : UU l3} (g h : X → A) → (g ＝ h) ≃ (f ∘ g ＝ f ∘ h)
+  equiv-postcomp-is-emb H {l3} = equiv-postcomp-is-mono l3 f (is-mono-is-emb H)
+
+equiv-postcomp-emb :
+  {l1 l2 l3 : Level}
+  {A : UU l1} {B : UU l2} {X : UU l3}
+  (f : A ↪ B) (g h : X → A) →
+  (g ＝ h) ≃ (map-emb f ∘ g ＝ map-emb f ∘ h)
+equiv-postcomp-emb (f , H) = equiv-postcomp-is-emb f H
 ```
 
 We construct an explicit equivalence for postcomposition for homotopies between
@@ -122,7 +134,7 @@ module _
       is-retraction-map-inv-is-equiv (pr2 f (g x) (h x)) (H x))
 
   equiv-htpy-postcomp-is-emb :
-    (g ~ h) ≃ ((pr1 f ∘ g) ~ (pr1 f ∘ h))
+    (g ~ h) ≃ (pr1 f ∘ g ~ pr1 f ∘ h)
   pr1 equiv-htpy-postcomp-is-emb = pr1 f ·l_
   pr2 equiv-htpy-postcomp-is-emb =
     is-equiv-is-invertible
diff --git a/src/foundation/subuniverses.lagda.md b/src/foundation/subuniverses.lagda.md
index ddeb1e28d6..c16067397a 100644
--- a/src/foundation/subuniverses.lagda.md
+++ b/src/foundation/subuniverses.lagda.md
@@ -89,20 +89,24 @@ module _
   is-essentially-in-subuniverse X =
     Σ (type-subuniverse P) (λ Y → inclusion-subuniverse P Y ≃ X)
 
+  is-proof-irrelevant-is-essentially-in-subuniverse :
+    {l3 : Level} (X : UU l3) → is-proof-irrelevant (is-essentially-in-subuniverse X)
+  is-proof-irrelevant-is-essentially-in-subuniverse X ((X' , p) , e) =
+    is-torsorial-Eq-subtype
+      ( is-contr-equiv'
+        ( Σ (UU _) (λ T → T ≃ X'))
+        ( equiv-tot (equiv-postcomp-equiv e))
+        ( is-torsorial-equiv' X'))
+      ( is-prop-is-in-subuniverse P)
+      ( X')
+      ( e)
+      ( p)
+
   is-prop-is-essentially-in-subuniverse :
     {l3 : Level} (X : UU l3) → is-prop (is-essentially-in-subuniverse X)
   is-prop-is-essentially-in-subuniverse X =
     is-prop-is-proof-irrelevant
-      ( λ ((X' , p) , e) →
-        is-torsorial-Eq-subtype
-          ( is-contr-equiv'
-            ( Σ (UU _) (λ T → T ≃ X'))
-            ( equiv-tot (equiv-postcomp-equiv e))
-            ( is-torsorial-equiv' X'))
-          ( is-prop-is-in-subuniverse P)
-          ( X')
-          ( e)
-          ( p))
+      ( is-proof-irrelevant-is-essentially-in-subuniverse X)
 
   is-essentially-in-subuniverse-Prop :
     {l3 : Level} (X : UU l3) → Prop (lsuc l1 ⊔ l2 ⊔ l3)
diff --git a/src/foundation/univalent-type-families.lagda.md b/src/foundation/univalent-type-families.lagda.md
index 9a7c109a9a..868ed207df 100644
--- a/src/foundation/univalent-type-families.lagda.md
+++ b/src/foundation/univalent-type-families.lagda.md
@@ -11,8 +11,15 @@ open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.equality-dependent-pair-types
 open import foundation.equivalences
+open import foundation.monomorphisms
+open import foundation.embeddings
+open import foundation.subtype-identity-principle
 open import foundation.fundamental-theorem-of-identity-types
 open import foundation.identity-systems
+open import foundation.identity-systems
+open import foundation.contractible-types
+open import foundation.functoriality-dependent-pair-types
+open import foundation.fibers-of-maps
 open import foundation.iterated-dependent-product-types
 open import foundation.propositions
 open import foundation.subuniverses
@@ -102,7 +109,7 @@ is-univalent-UU l = univalence
 ```
 
 where the right edge is an equivalence by the univalence axiom. Hence, the top
-map is an equivalence if and only if the left map is.
+map is an equivalence if and only if the left map is. ∎
 
 ```agda
 module _
@@ -169,6 +176,42 @@ module _
     is-univalent-is-emb (is-emb-inclusion-subuniverse S)
 ```
 
+### The underlying subuniverse of a univalent type family
+
+```agda
+module _
+  {l1 l2 : Level} {A : UU l1} ((B , H) : univalent-type-family l2 A)
+  where
+
+  emb-univalent-type-family : A ↪ UU l2
+  emb-univalent-type-family = (B , is-emb-is-univalent H)
+
+  is-in-subuniverse-univalent-family : {l3 : Level} → UU l3 → UU (l1 ⊔ l2 ⊔ l3)
+  is-in-subuniverse-univalent-family X = Σ A (λ a → (B a ≃ X))
+
+  abstract
+    is-prop-is-in-subuniverse-univalent-family :
+      {l3 : Level} {X : UU l3} → is-prop (is-in-subuniverse-univalent-family X)
+    is-prop-is-in-subuniverse-univalent-family {X = X} =
+      is-prop-emb
+        ( emb-Σ-emb-base emb-univalent-type-family (λ Y → Y ≃ X))
+        ( is-prop-is-proof-irrelevant
+          ( λ (Y , e) →
+            is-contr-equiv'
+              ( Σ (UU l2) (λ Y' → Y' ≃ Y))
+              ( equiv-tot (equiv-postcomp-equiv e))
+              ( is-torsorial-equiv' Y)))
+
+  is-in-subuniverse-prop-univalent-family :
+    {l3 : Level} → UU l3 → Prop (l1 ⊔ l2 ⊔ l3)
+  is-in-subuniverse-prop-univalent-family X =
+    ( is-in-subuniverse-univalent-family X ,
+      is-prop-is-in-subuniverse-univalent-family)
+
+  subuniverse-univalent-family : (l3 : Level) → subuniverse l3 (l1 ⊔ l2 ⊔ l3)
+  subuniverse-univalent-family l3 = is-in-subuniverse-prop-univalent-family
+```
+
 ## See also
 
 - [Preunivalent type families](foundation.preunivalent-type-families.md)

From 7fbc0f814d21c502b917c9f08523a11054726c10 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 13 Sep 2025 15:25:46 +0200
Subject: [PATCH 13/41] univalent polynomial endofunctors

---
 .../univalent-type-families.lagda.md          |  53 +++-
 src/trees.lagda.md                            |   1 +
 src/trees/polynomial-endofunctors.lagda.md    |  41 +--
 ...univalent-polynomial-endofunctors.lagda.md | 262 ++++++++++++++++++
 4 files changed, 329 insertions(+), 28 deletions(-)
 create mode 100644 src/trees/univalent-polynomial-endofunctors.lagda.md

diff --git a/src/foundation/univalent-type-families.lagda.md b/src/foundation/univalent-type-families.lagda.md
index 868ed207df..f4da273881 100644
--- a/src/foundation/univalent-type-families.lagda.md
+++ b/src/foundation/univalent-type-families.lagda.md
@@ -8,27 +8,26 @@ module foundation.univalent-type-families where
 
 ```agda
 open import foundation.action-on-identifications-functions
+open import foundation.contractible-types
 open import foundation.dependent-pair-types
+open import foundation.embeddings
 open import foundation.equality-dependent-pair-types
 open import foundation.equivalences
-open import foundation.monomorphisms
-open import foundation.embeddings
-open import foundation.subtype-identity-principle
+open import foundation.fibers-of-maps
+open import foundation.functoriality-dependent-pair-types
 open import foundation.fundamental-theorem-of-identity-types
+open import foundation.global-subuniverses
 open import foundation.identity-systems
-open import foundation.identity-systems
-open import foundation.contractible-types
-open import foundation.functoriality-dependent-pair-types
-open import foundation.fibers-of-maps
 open import foundation.iterated-dependent-product-types
+open import foundation.monomorphisms
 open import foundation.propositions
+open import foundation.subtype-identity-principle
 open import foundation.subuniverses
 open import foundation.transport-along-identifications
 open import foundation.univalence
 open import foundation.universal-property-identity-systems
 open import foundation.universe-levels
 
-open import foundation-core.embeddings
 open import foundation-core.function-types
 open import foundation-core.identity-types
 open import foundation-core.sections
@@ -81,6 +80,26 @@ module _
 univalent-type-family :
   {l1 : Level} (l2 : Level) (A : UU l1) → UU (l1 ⊔ lsuc l2)
 univalent-type-family l2 A = Σ (A → UU l2) is-univalent
+
+module _
+  {l1 l2 : Level} {A : UU l1} (ℬ : univalent-type-family l2 A)
+  where
+
+  type-family-univalent-type-family : A → UU l2
+  type-family-univalent-type-family = pr1 ℬ
+
+  is-univalent-univalent-type-family :
+    is-univalent type-family-univalent-type-family
+  is-univalent-univalent-type-family =
+    pr2 ℬ
+
+  equiv-equiv-tr-univalent-type-family :
+    {x y : A} →
+    ( x ＝ y) ≃
+    ( type-family-univalent-type-family x ≃ type-family-univalent-type-family y)
+  equiv-equiv-tr-univalent-type-family {x} {y} =
+    ( equiv-tr type-family-univalent-type-family ,
+      is-univalent-univalent-type-family x y)
 ```
 
 ## Properties
@@ -176,7 +195,7 @@ module _
     is-univalent-is-emb (is-emb-inclusion-subuniverse S)
 ```
 
-### The underlying subuniverse of a univalent type family
+### The underlying global subuniverse of a univalent type family
 
 ```agda
 module _
@@ -210,6 +229,22 @@ module _
 
   subuniverse-univalent-family : (l3 : Level) → subuniverse l3 (l1 ⊔ l2 ⊔ l3)
   subuniverse-univalent-family l3 = is-in-subuniverse-prop-univalent-family
+
+  is-closed-under-equiv-subuniverse-univalent-family :
+    {l3 l4 : Level} →
+    is-closed-under-equiv-subuniverses
+      (λ l → l1 ⊔ l2 ⊔ l) subuniverse-univalent-family l3 l4
+  is-closed-under-equiv-subuniverse-univalent-family X Y f (a , e) =
+    ( a , f ∘e e)
+
+  global-subuniverse-univalent-family :
+    global-subuniverse (λ l → l1 ⊔ l2 ⊔ l)
+  global-subuniverse-univalent-family =
+    λ where
+      .subuniverse-global-subuniverse →
+        subuniverse-univalent-family
+      .is-closed-under-equiv-global-subuniverse →
+        is-closed-under-equiv-subuniverse-univalent-family
 ```
 
 ## See also
diff --git a/src/trees.lagda.md b/src/trees.lagda.md
index 5d2a613423..55c954262b 100644
--- a/src/trees.lagda.md
+++ b/src/trees.lagda.md
@@ -64,6 +64,7 @@ open import trees.transitive-multisets public
 open import trees.underlying-trees-elements-coalgebras-polynomial-endofunctors public
 open import trees.underlying-trees-of-elements-of-w-types public
 open import trees.undirected-trees public
+open import trees.univalent-polynomial-endofunctors public
 open import trees.universal-multiset public
 open import trees.universal-tree public
 open import trees.w-type-of-natural-numbers public
diff --git a/src/trees/polynomial-endofunctors.lagda.md b/src/trees/polynomial-endofunctors.lagda.md
index 6139f5a1b8..044cec1a61 100644
--- a/src/trees/polynomial-endofunctors.lagda.md
+++ b/src/trees/polynomial-endofunctors.lagda.md
@@ -33,7 +33,8 @@ open import foundation-core.torsorial-type-families
 ## Idea
 
 Given a type `A` [equipped](foundation.structure.md) with a type family `B` over
-`A`, the {{#concept "polynomial endofunctor" Agda=polynomial-endofunctor}}
+`A`, the
+{{#concept "polynomial endofunctor" WD="polynomial functor" WDID=Q48842893 Agda=polynomial-endofunctor}}
 `𝑃 A B` is defined by
 
 ```text
@@ -80,7 +81,25 @@ type-polynomial-endofunctor :
 type-polynomial-endofunctor (A , B) = type-polynomial-endofunctor' A B
 ```
 
-### The identity type of `type-polynomial-endofunctor`
+### The action on maps of the polynomial endofunctor
+
+```agda
+map-polynomial-endofunctor' :
+  {l1 l2 l3 l4 : Level} (A : UU l1) (B : A → UU l2) {X : UU l3} {Y : UU l4}
+  (f : X → Y) →
+  type-polynomial-endofunctor' A B X → type-polynomial-endofunctor' A B Y
+map-polynomial-endofunctor' A B f = tot (λ x α → f ∘ α)
+
+map-polynomial-endofunctor :
+  {l1 l2 l3 l4 : Level} (𝑃 : polynomial-endofunctor l1 l2)
+  {X : UU l3} {Y : UU l4} (f : X → Y) →
+  type-polynomial-endofunctor 𝑃 X → type-polynomial-endofunctor 𝑃 Y
+map-polynomial-endofunctor (A , B) = map-polynomial-endofunctor' A B
+```
+
+## Properties
+
+### Characterizing identity in the image of polynomial endofunctors
 
 ```agda
 module _
@@ -143,22 +162,6 @@ module _
     is-retraction-eq-Eq-type-polynomial-endofunctor x x refl
 ```
 
-### The action on maps of the polynomial endofunctor
-
-```agda
-map-polynomial-endofunctor' :
-  {l1 l2 l3 l4 : Level} (A : UU l1) (B : A → UU l2) {X : UU l3} {Y : UU l4}
-  (f : X → Y) →
-  type-polynomial-endofunctor' A B X → type-polynomial-endofunctor' A B Y
-map-polynomial-endofunctor' A B f = tot (λ x α → f ∘ α)
-
-map-polynomial-endofunctor :
-  {l1 l2 l3 l4 : Level} (𝑃 : polynomial-endofunctor l1 l2)
-  {X : UU l3} {Y : UU l4} (f : X → Y) →
-  type-polynomial-endofunctor 𝑃 X → type-polynomial-endofunctor 𝑃 Y
-map-polynomial-endofunctor (A , B) = map-polynomial-endofunctor' A B
-```
-
 ### The action on homotopies of the polynomial endofunctor
 
 ```agda
@@ -194,7 +197,7 @@ coh-refl-htpy-polynomial-endofunctor (A , B) =
   coh-refl-htpy-polynomial-endofunctor' A B
 ```
 
-### Computing the fiber of the action on maps
+### Computing the fibers of the action on maps
 
 ```agda
 module _
diff --git a/src/trees/univalent-polynomial-endofunctors.lagda.md b/src/trees/univalent-polynomial-endofunctors.lagda.md
new file mode 100644
index 0000000000..b0ae7563a0
--- /dev/null
+++ b/src/trees/univalent-polynomial-endofunctors.lagda.md
@@ -0,0 +1,262 @@
+# Univalent polynomial endofunctors
+
+```agda
+module trees.univalent-polynomial-endofunctors where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.contractible-types
+open import foundation.dependent-pair-types
+open import foundation.equivalence-induction
+open import foundation.equivalences
+open import foundation.fibers-of-maps
+open import foundation.function-types
+open import foundation.functoriality-dependent-pair-types
+open import foundation.fundamental-theorem-of-identity-types
+open import foundation.global-subuniverses
+open import foundation.homotopies
+open import foundation.homotopy-induction
+open import foundation.identity-types
+open import foundation.locally-small-types
+open import foundation.postcomposition-functions
+open import foundation.propositions
+open import foundation.structure-identity-principle
+open import foundation.subuniverses
+open import foundation.transport-along-identifications
+open import foundation.univalence
+open import foundation.univalent-type-families
+open import foundation.universe-levels
+open import foundation.whiskering-homotopies-composition
+
+open import foundation-core.commuting-triangles-of-maps
+open import foundation-core.retractions
+open import foundation-core.torsorial-type-families
+
+open import trees.polynomial-endofunctors
+```
+
+</details>
+
+## Idea
+
+A [polynomial endofunctor](trees.polynomial-endofunctors.md) `𝑃` is
+{{#concept "univalent" Disambiguation="polynomial endofunctor" Agda=is-univalent-polynomial-endofunctor Agda=univalent-polynomial-endofunctor}}
+if its type family of positions is
+[univalent](foundation.univalent-type-families.md). In other words, a polynomial
+endofunctor is univalent if it is equivalent to the polynomial endofunctor given
+by a [subuniverse](foundation.subuniverses.md)
+
+```text
+  𝑃 ≃ (𝒮 ◃ pr1).
+```
+
+## Definitions
+
+### The predicate on polynomial endofunctors of being univalent
+
+```agda
+is-univalent-polynomial-endofunctor :
+  {l1 l2 : Level} → polynomial-endofunctor l1 l2 → UU (l1 ⊔ l2)
+is-univalent-polynomial-endofunctor (A , B) = is-univalent B
+
+is-prop-is-univalent-polynomial-endofunctor :
+  {l1 l2 : Level} (𝑃 : polynomial-endofunctor l1 l2) →
+  is-prop (is-univalent-polynomial-endofunctor 𝑃)
+is-prop-is-univalent-polynomial-endofunctor 𝑃 = is-prop-is-univalent
+
+is-univalent-polynomial-endofunctor-Prop :
+  {l1 l2 : Level} → polynomial-endofunctor l1 l2 → Prop (l1 ⊔ l2)
+is-univalent-polynomial-endofunctor-Prop 𝑃 =
+  ( is-univalent-polynomial-endofunctor 𝑃 ,
+    is-prop-is-univalent-polynomial-endofunctor 𝑃)
+```
+
+### The type of univalent polynomial endofunctors
+
+```agda
+univalent-polynomial-endofunctor : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)
+univalent-polynomial-endofunctor l1 l2 =
+  Σ (polynomial-endofunctor l1 l2) (is-univalent-polynomial-endofunctor)
+
+module _
+  {l1 l2 : Level} (𝑃 : univalent-polynomial-endofunctor l1 l2)
+  where
+
+  polynomial-endofunctor-univalent-polynomial-endofunctor :
+    polynomial-endofunctor l1 l2
+  polynomial-endofunctor-univalent-polynomial-endofunctor = pr1 𝑃
+
+  shapes-univalent-polynomial-endofunctor : UU l1
+  shapes-univalent-polynomial-endofunctor =
+    shapes-polynomial-endofunctor
+      polynomial-endofunctor-univalent-polynomial-endofunctor
+
+  positions-univalent-polynomial-endofunctor :
+    shapes-univalent-polynomial-endofunctor → UU l2
+  positions-univalent-polynomial-endofunctor =
+    positions-polynomial-endofunctor
+      polynomial-endofunctor-univalent-polynomial-endofunctor
+
+  is-univalent-univalent-polynomial-endofunctor :
+    is-univalent-polynomial-endofunctor
+      polynomial-endofunctor-univalent-polynomial-endofunctor
+  is-univalent-univalent-polynomial-endofunctor = pr2 𝑃
+
+  univalent-family-univalent-polynomial-endofunctor :
+    univalent-type-family l2 shapes-univalent-polynomial-endofunctor
+  univalent-family-univalent-polynomial-endofunctor =
+    ( positions-univalent-polynomial-endofunctor ,
+      is-univalent-univalent-polynomial-endofunctor)
+
+  equiv-equiv-tr-univalent-polynomial-endofunctor :
+    {x y : shapes-univalent-polynomial-endofunctor} →
+    ( x ＝ y) ≃
+    ( positions-univalent-polynomial-endofunctor x ≃
+      positions-univalent-polynomial-endofunctor y)
+  equiv-equiv-tr-univalent-polynomial-endofunctor =
+    equiv-equiv-tr-univalent-type-family
+      univalent-family-univalent-polynomial-endofunctor
+
+make-univalent-polynomial-endofunctor :
+  {l1 l2 : Level} {A : UU l1} (B : A → UU l2) → is-univalent B →
+  univalent-polynomial-endofunctor l1 l2
+make-univalent-polynomial-endofunctor B H = ((_ , B) , H)
+```
+
+### The underlying subuniverse of a univalent polynomial endofunctor
+
+```agda
+module _
+  {l1 l2 : Level} (𝑃 : univalent-polynomial-endofunctor l1 l2)
+  where
+
+  subuniverse-univalent-polynomial-endofunctor :
+    (l3 : Level) → subuniverse l3 (l1 ⊔ l2 ⊔ l3)
+  subuniverse-univalent-polynomial-endofunctor =
+    subuniverse-univalent-family
+      ( univalent-family-univalent-polynomial-endofunctor 𝑃)
+
+  global-subuniverse-univalent-polynomial-endofunctor :
+    global-subuniverse (λ l3 → l1 ⊔ l2 ⊔ l3)
+  global-subuniverse-univalent-polynomial-endofunctor =
+    global-subuniverse-univalent-family
+      ( univalent-family-univalent-polynomial-endofunctor 𝑃)
+```
+
+### The action on types of a univalent polynomial endofunctor
+
+```agda
+type-univalent-polynomial-endofunctor :
+  {l1 l2 l3 : Level} →
+  univalent-polynomial-endofunctor l1 l2 →
+  UU l3 → UU (l1 ⊔ l2 ⊔ l3)
+type-univalent-polynomial-endofunctor 𝑃 =
+  type-polynomial-endofunctor
+    ( polynomial-endofunctor-univalent-polynomial-endofunctor 𝑃)
+```
+
+### The action on maps of a univalent polynomial endofunctor
+
+```agda
+map-univalent-polynomial-endofunctor :
+  {l1 l2 l3 l4 : Level} (𝑃 : univalent-polynomial-endofunctor l1 l2)
+  {X : UU l3} {Y : UU l4} (f : X → Y) →
+  type-univalent-polynomial-endofunctor 𝑃 X →
+  type-univalent-polynomial-endofunctor 𝑃 Y
+map-univalent-polynomial-endofunctor 𝑃 =
+  map-polynomial-endofunctor
+    ( polynomial-endofunctor-univalent-polynomial-endofunctor 𝑃)
+```
+
+### Characterizing identity in the image of univalent polynomial endofunctors
+
+```agda
+module _
+  {l1 l2 l3 : Level} (𝑃 : univalent-polynomial-endofunctor l1 l2) {X : UU l3}
+  where
+
+  Eq-type-univalent-polynomial-endofunctor :
+    (x y : type-univalent-polynomial-endofunctor 𝑃 X) → UU (l2 ⊔ l3)
+  Eq-type-univalent-polynomial-endofunctor x y =
+    Σ ( positions-univalent-polynomial-endofunctor 𝑃 (pr1 x) ≃
+        positions-univalent-polynomial-endofunctor 𝑃 (pr1 y))
+      ( λ e → coherence-triangle-maps (pr2 x) (pr2 y) (map-equiv e))
+
+  refl-Eq-type-univalent-polynomial-endofunctor :
+    (x : type-univalent-polynomial-endofunctor 𝑃 X) →
+    Eq-type-univalent-polynomial-endofunctor x x
+  refl-Eq-type-univalent-polynomial-endofunctor (x , α) = (id-equiv , refl-htpy)
+
+  Eq-eq-type-univalent-polynomial-endofunctor :
+    (x y : type-univalent-polynomial-endofunctor 𝑃 X) →
+    x ＝ y → Eq-type-univalent-polynomial-endofunctor x y
+  Eq-eq-type-univalent-polynomial-endofunctor x .x refl =
+    refl-Eq-type-univalent-polynomial-endofunctor x
+
+  abstract
+    is-torsorial-Eq-type-univalent-polynomial-endofunctor :
+      (x : type-univalent-polynomial-endofunctor 𝑃 X) →
+      is-torsorial (Eq-type-univalent-polynomial-endofunctor x)
+    is-torsorial-Eq-type-univalent-polynomial-endofunctor (x , α) =
+      is-torsorial-Eq-structure
+        ( is-contr-equiv'
+          ( Σ (shapes-univalent-polynomial-endofunctor 𝑃) (x ＝_))
+          ( equiv-tot
+            ( λ y → equiv-equiv-tr-univalent-polynomial-endofunctor 𝑃 {x} {y}))
+          ( is-torsorial-Id x))
+        ( x , id-equiv)
+        ( is-torsorial-htpy α)
+
+  is-equiv-Eq-eq-type-univalent-polynomial-endofunctor :
+    (x y : type-univalent-polynomial-endofunctor 𝑃 X) →
+    is-equiv (Eq-eq-type-univalent-polynomial-endofunctor x y)
+  is-equiv-Eq-eq-type-univalent-polynomial-endofunctor x =
+    fundamental-theorem-id
+      ( is-torsorial-Eq-type-univalent-polynomial-endofunctor x)
+      ( Eq-eq-type-univalent-polynomial-endofunctor x)
+
+  extensionality-type-univalent-polynomial-endofunctor :
+    (x y : type-univalent-polynomial-endofunctor 𝑃 X) →
+    (x ＝ y) ≃ Eq-type-univalent-polynomial-endofunctor x y
+  extensionality-type-univalent-polynomial-endofunctor x y =
+    ( Eq-eq-type-univalent-polynomial-endofunctor x y ,
+      is-equiv-Eq-eq-type-univalent-polynomial-endofunctor x y)
+
+  eq-Eq-type-univalent-polynomial-endofunctor :
+    (x y : type-univalent-polynomial-endofunctor 𝑃 X) →
+    Eq-type-univalent-polynomial-endofunctor x y → x ＝ y
+  eq-Eq-type-univalent-polynomial-endofunctor x y =
+    map-inv-equiv (extensionality-type-univalent-polynomial-endofunctor x y)
+
+  is-retraction-eq-Eq-type-univalent-polynomial-endofunctor :
+    (x y : type-univalent-polynomial-endofunctor 𝑃 X) →
+    is-retraction
+      ( Eq-eq-type-univalent-polynomial-endofunctor x y)
+      ( eq-Eq-type-univalent-polynomial-endofunctor x y)
+  is-retraction-eq-Eq-type-univalent-polynomial-endofunctor x y =
+    is-retraction-map-inv-is-equiv
+      ( is-equiv-Eq-eq-type-univalent-polynomial-endofunctor x y)
+
+  coh-refl-eq-Eq-type-univalent-polynomial-endofunctor :
+    (x : type-univalent-polynomial-endofunctor 𝑃 X) →
+    ( eq-Eq-type-univalent-polynomial-endofunctor x x
+      ( refl-Eq-type-univalent-polynomial-endofunctor x)) ＝ refl
+  coh-refl-eq-Eq-type-univalent-polynomial-endofunctor x =
+    is-retraction-eq-Eq-type-univalent-polynomial-endofunctor x x refl
+```
+
+### Local smallness of the image of univalent endofunctors
+
+```agda
+module _
+  {l1 l2 l3 : Level} (𝑃 : univalent-polynomial-endofunctor l1 l2) {X : UU l3}
+  where
+
+  is-locally-small-type-univalent-polynomial-endofunctor :
+    is-locally-small (l2 ⊔ l3) (type-univalent-polynomial-endofunctor 𝑃 X)
+  is-locally-small-type-univalent-polynomial-endofunctor x y =
+    ( Eq-type-univalent-polynomial-endofunctor 𝑃 x y ,
+      extensionality-type-univalent-polynomial-endofunctor 𝑃 x y)
+```

From 7db557284d4d75a82fb635ee813e7a44fa591cff Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 13 Sep 2025 17:30:08 +0200
Subject: [PATCH 14/41] The indexing type of a univalent type family is
 equivalent to the subuniverse

---
 .../univalent-type-families.lagda.md          | 96 +++++++++++++++----
 1 file changed, 78 insertions(+), 18 deletions(-)

diff --git a/src/foundation/univalent-type-families.lagda.md b/src/foundation/univalent-type-families.lagda.md
index f4da273881..281caa5fa9 100644
--- a/src/foundation/univalent-type-families.lagda.md
+++ b/src/foundation/univalent-type-families.lagda.md
@@ -17,6 +17,7 @@ open import foundation.fibers-of-maps
 open import foundation.functoriality-dependent-pair-types
 open import foundation.fundamental-theorem-of-identity-types
 open import foundation.global-subuniverses
+open import foundation.retractions
 open import foundation.identity-systems
 open import foundation.iterated-dependent-product-types
 open import foundation.monomorphisms
@@ -77,29 +78,29 @@ module _
 ### Univalent type families
 
 ```agda
-univalent-type-family :
+univalent-family :
   {l1 : Level} (l2 : Level) (A : UU l1) → UU (l1 ⊔ lsuc l2)
-univalent-type-family l2 A = Σ (A → UU l2) is-univalent
+univalent-family l2 A = Σ (A → UU l2) is-univalent
 
 module _
-  {l1 l2 : Level} {A : UU l1} (ℬ : univalent-type-family l2 A)
+  {l1 l2 : Level} {A : UU l1} (ℬ : univalent-family l2 A)
   where
 
-  type-family-univalent-type-family : A → UU l2
-  type-family-univalent-type-family = pr1 ℬ
+  type-family-univalent-family : A → UU l2
+  type-family-univalent-family = pr1 ℬ
 
-  is-univalent-univalent-type-family :
-    is-univalent type-family-univalent-type-family
-  is-univalent-univalent-type-family =
+  is-univalent-univalent-family :
+    is-univalent type-family-univalent-family
+  is-univalent-univalent-family =
     pr2 ℬ
 
-  equiv-equiv-tr-univalent-type-family :
+  equiv-equiv-tr-univalent-family :
     {x y : A} →
     ( x ＝ y) ≃
-    ( type-family-univalent-type-family x ≃ type-family-univalent-type-family y)
-  equiv-equiv-tr-univalent-type-family {x} {y} =
-    ( equiv-tr type-family-univalent-type-family ,
-      is-univalent-univalent-type-family x y)
+    ( type-family-univalent-family x ≃ type-family-univalent-family y)
+  equiv-equiv-tr-univalent-family {x} {y} =
+    ( equiv-tr type-family-univalent-family ,
+      is-univalent-univalent-family x y)
 ```
 
 ## Properties
@@ -157,6 +158,10 @@ module _
         ( λ where refl → refl)
         ( is-emb-B x y)
         ( univalence (B x) (B y))
+
+emb-univalent-family :
+  {l1 l2 : Level} {A : UU l1} → univalent-family l2 A → A ↪ UU l2
+emb-univalent-family (B , H) = (B , is-emb-is-univalent H)
 ```
 
 ### Univalent type families satisfy equivalence induction
@@ -199,12 +204,9 @@ module _
 
 ```agda
 module _
-  {l1 l2 : Level} {A : UU l1} ((B , H) : univalent-type-family l2 A)
+  {l1 l2 : Level} {A : UU l1} ((B , H) : univalent-family l2 A)
   where
 
-  emb-univalent-type-family : A ↪ UU l2
-  emb-univalent-type-family = (B , is-emb-is-univalent H)
-
   is-in-subuniverse-univalent-family : {l3 : Level} → UU l3 → UU (l1 ⊔ l2 ⊔ l3)
   is-in-subuniverse-univalent-family X = Σ A (λ a → (B a ≃ X))
 
@@ -213,7 +215,7 @@ module _
       {l3 : Level} {X : UU l3} → is-prop (is-in-subuniverse-univalent-family X)
     is-prop-is-in-subuniverse-univalent-family {X = X} =
       is-prop-emb
-        ( emb-Σ-emb-base emb-univalent-type-family (λ Y → Y ≃ X))
+        ( emb-Σ-emb-base (emb-univalent-family (B , H)) (λ Y → Y ≃ X))
         ( is-prop-is-proof-irrelevant
           ( λ (Y , e) →
             is-contr-equiv'
@@ -247,6 +249,64 @@ module _
         is-closed-under-equiv-subuniverse-univalent-family
 ```
 
+### The indexing type of a univalent type family is equivalent to the subuniverse
+
+```agda
+module _
+  {l1 l2 : Level} {A : UU l1} (ℬ : univalent-family l2 A)
+  where
+
+  map-equiv-type-subuniverse-univalent-family :
+    A → type-subuniverse (subuniverse-univalent-family ℬ l2)
+  map-equiv-type-subuniverse-univalent-family a =
+    (type-family-univalent-family ℬ a , a , id-equiv)
+
+  map-inv-equiv-type-subuniverse-univalent-family :
+     type-subuniverse (subuniverse-univalent-family ℬ l2) → A
+  map-inv-equiv-type-subuniverse-univalent-family (X , a , e) = a
+
+  is-retraction-map-inv-equiv-type-subuniverse-univalent-family :
+    is-retraction
+      map-equiv-type-subuniverse-univalent-family
+      map-inv-equiv-type-subuniverse-univalent-family
+  is-retraction-map-inv-equiv-type-subuniverse-univalent-family a = refl
+
+  is-section-map-inv-equiv-type-subuniverse-univalent-family :
+    is-section
+      map-equiv-type-subuniverse-univalent-family
+      map-inv-equiv-type-subuniverse-univalent-family
+  is-section-map-inv-equiv-type-subuniverse-univalent-family (X , a , e) =
+    eq-equiv-subuniverse (subuniverse-univalent-family ℬ l2) e
+
+  is-equiv-map-equiv-type-subuniverse-univalent-family :
+    is-equiv map-equiv-type-subuniverse-univalent-family
+  is-equiv-map-equiv-type-subuniverse-univalent-family =
+    is-equiv-is-invertible
+      map-inv-equiv-type-subuniverse-univalent-family
+      is-section-map-inv-equiv-type-subuniverse-univalent-family
+      is-retraction-map-inv-equiv-type-subuniverse-univalent-family
+
+  is-equiv-map-inv-equiv-type-subuniverse-univalent-family :
+    is-equiv map-inv-equiv-type-subuniverse-univalent-family
+  is-equiv-map-inv-equiv-type-subuniverse-univalent-family =
+    is-equiv-is-invertible
+      map-equiv-type-subuniverse-univalent-family
+      is-retraction-map-inv-equiv-type-subuniverse-univalent-family
+      is-section-map-inv-equiv-type-subuniverse-univalent-family
+
+  equiv-type-subuniverse-univalent-family :
+    A ≃ type-subuniverse (subuniverse-univalent-family ℬ l2)
+  equiv-type-subuniverse-univalent-family =
+    ( map-equiv-type-subuniverse-univalent-family ,
+      is-equiv-map-equiv-type-subuniverse-univalent-family)
+
+  inv-equiv-type-subuniverse-univalent-family :
+    type-subuniverse (subuniverse-univalent-family ℬ l2) ≃ A
+  inv-equiv-type-subuniverse-univalent-family =
+    ( map-inv-equiv-type-subuniverse-univalent-family ,
+      is-equiv-map-inv-equiv-type-subuniverse-univalent-family)
+```
+
 ## See also
 
 - [Preunivalent type families](foundation.preunivalent-type-families.md)

From 601be76663a00487dc72d64eb51f8c4c370a9726 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 13 Sep 2025 17:30:21 +0200
Subject: [PATCH 15/41] shorten to `univalent-family`

---
 src/trees/univalent-polynomial-endofunctors.lagda.md | 9 +++++++--
 1 file changed, 7 insertions(+), 2 deletions(-)

diff --git a/src/trees/univalent-polynomial-endofunctors.lagda.md b/src/trees/univalent-polynomial-endofunctors.lagda.md
index b0ae7563a0..aac0e4c1d4 100644
--- a/src/trees/univalent-polynomial-endofunctors.lagda.md
+++ b/src/trees/univalent-polynomial-endofunctors.lagda.md
@@ -105,7 +105,7 @@ module _
   is-univalent-univalent-polynomial-endofunctor = pr2 𝑃
 
   univalent-family-univalent-polynomial-endofunctor :
-    univalent-type-family l2 shapes-univalent-polynomial-endofunctor
+    univalent-family l2 shapes-univalent-polynomial-endofunctor
   univalent-family-univalent-polynomial-endofunctor =
     ( positions-univalent-polynomial-endofunctor ,
       is-univalent-univalent-polynomial-endofunctor)
@@ -116,7 +116,7 @@ module _
     ( positions-univalent-polynomial-endofunctor x ≃
       positions-univalent-polynomial-endofunctor y)
   equiv-equiv-tr-univalent-polynomial-endofunctor =
-    equiv-equiv-tr-univalent-type-family
+    equiv-equiv-tr-univalent-family
       univalent-family-univalent-polynomial-endofunctor
 
 make-univalent-polynomial-endofunctor :
@@ -138,6 +138,11 @@ module _
     subuniverse-univalent-family
       ( univalent-family-univalent-polynomial-endofunctor 𝑃)
 
+  is-in-subuniverse-univalent-polynomial-endofunctor :
+    {l3 : Level} → UU l3 → UU (l1 ⊔ l2 ⊔ l3)
+  is-in-subuniverse-univalent-polynomial-endofunctor {l3} =
+    is-in-subuniverse (subuniverse-univalent-polynomial-endofunctor l3)
+
   global-subuniverse-univalent-polynomial-endofunctor :
     global-subuniverse (λ l3 → l1 ⊔ l2 ⊔ l3)
   global-subuniverse-univalent-polynomial-endofunctor =

From cc7c9c59ebeb99bb238564e4f8f49fe90e114900 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 13 Sep 2025 18:01:41 +0200
Subject: [PATCH 16/41] add some "see also"s

---
 src/foundation-core/small-types.lagda.md | 2 ++
 src/foundation/small-types.lagda.md      | 2 ++
 src/foundation/subuniverses.lagda.md     | 2 ++
 3 files changed, 6 insertions(+)

diff --git a/src/foundation-core/small-types.lagda.md b/src/foundation-core/small-types.lagda.md
index 838cd07547..46f8711914 100644
--- a/src/foundation-core/small-types.lagda.md
+++ b/src/foundation-core/small-types.lagda.md
@@ -353,3 +353,5 @@ is-small-logical-equivalence H K =
 ## See also
 
 - [Small maps](foundation.small-maps.md)
+- The `is-essentially-in-subuniverse` predicate in
+  [`foundation.subuniverses`](foundation.subuniverses.md)
diff --git a/src/foundation/small-types.lagda.md b/src/foundation/small-types.lagda.md
index 2d60593a75..45497899e8 100644
--- a/src/foundation/small-types.lagda.md
+++ b/src/foundation/small-types.lagda.md
@@ -45,3 +45,5 @@ is-small-is-surjective {f = f} H K L =
 ## See also
 
 - [Small maps](foundation.small-maps.md)
+- The `is-essentially-in-subuniverse` predicate in
+  [`foundation.subuniverses`](foundation.subuniverses.md)
diff --git a/src/foundation/subuniverses.lagda.md b/src/foundation/subuniverses.lagda.md
index c16067397a..2a19ec8bfd 100644
--- a/src/foundation/subuniverses.lagda.md
+++ b/src/foundation/subuniverses.lagda.md
@@ -116,6 +116,8 @@ module _
     is-prop-is-essentially-in-subuniverse X
 ```
 
+- See also [univalent type families](foundation.univalent-type-families.md).
+
 ## Properties
 
 ### Subuniverses are closed under equivalences

From 54c4cad082107ab050c0ff1b925722c0eb4ee58e Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 13 Sep 2025 18:18:59 +0200
Subject: [PATCH 17/41] edits

---
 .../univalent-type-families.lagda.md          |  2 +-
 ...morphisms-polynomial-endofunctors.lagda.md | 71 +++++++------------
 ...ormations-polynomial-endofunctors.lagda.md | 10 +++
 ...morphisms-polynomial-endofunctors.lagda.md | 29 ++------
 4 files changed, 42 insertions(+), 70 deletions(-)

diff --git a/src/foundation/univalent-type-families.lagda.md b/src/foundation/univalent-type-families.lagda.md
index 281caa5fa9..1d11324a68 100644
--- a/src/foundation/univalent-type-families.lagda.md
+++ b/src/foundation/univalent-type-families.lagda.md
@@ -17,11 +17,11 @@ open import foundation.fibers-of-maps
 open import foundation.functoriality-dependent-pair-types
 open import foundation.fundamental-theorem-of-identity-types
 open import foundation.global-subuniverses
-open import foundation.retractions
 open import foundation.identity-systems
 open import foundation.iterated-dependent-product-types
 open import foundation.monomorphisms
 open import foundation.propositions
+open import foundation.retractions
 open import foundation.subtype-identity-principle
 open import foundation.subuniverses
 open import foundation.transport-along-identifications
diff --git a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
index 7df96d5e4d..a33bd8b4df 100644
--- a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
@@ -20,6 +20,7 @@ open import foundation.equivalences
 open import foundation.equivalences-arrows
 open import foundation.fibers-of-maps
 open import foundation.function-types
+open import foundation.functoriality-dependent-function-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.functoriality-fibers-of-maps
 open import foundation.fundamental-theorem-of-identity-types
@@ -227,9 +228,10 @@ comp-cartesian-hom-polynomial-endofunctor :
   cartesian-hom-polynomial-endofunctor 𝑄 𝑅 →
   cartesian-hom-polynomial-endofunctor 𝑃 𝑄 →
   cartesian-hom-polynomial-endofunctor 𝑃 𝑅
-comp-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 𝑅 (β , H) (α , K) =
-  ( comp-hom-polynomial-endofunctor 𝑃 𝑄 𝑅 β α ,
-    λ a → is-equiv-comp (pr2 α a) (pr2 β (pr1 α a)) (H (pr1 α a)) (K a))
+comp-cartesian-hom-polynomial-endofunctor
+  𝑃 𝑄 𝑅 ((β₀ , β₁) , H) ((α₀ , α₁) , K) =
+  ( ( comp-hom-polynomial-endofunctor 𝑃 𝑄 𝑅 (β₀ , β₁) (α₀ , α₁)) ,
+    ( λ a → is-equiv-comp (α₁ a) (β₁ (α₀ a)) (H (α₀ a)) (K a)))
 ```
 
 ## Properties
@@ -380,59 +382,40 @@ module _
 
 ### Cartesian natural transformations define cartesian morphisms
 
-```text
+```agda
 module _
   {l1 l2 l3 l4 : Level}
   (𝑃 : polynomial-endofunctor l1 l2)
   (𝑄 : polynomial-endofunctor l3 l4)
-  (α : natural-transformation-polynomial-endofunctor l2 𝑃 𝑄)
+  (α : cartesian-natural-transformation-polynomial-endofunctor l2 𝑃 𝑄)
+  (let 𝑃₀ = shapes-polynomial-endofunctor 𝑃)
+  (let 𝑃₁ = positions-polynomial-endofunctor 𝑃)
+  (let α₀ = type-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α)
   where
 
-  shapes-natural-transformation-polynomial-endofunctor :
+  shapes-cartesian-natural-transformation-polynomial-endofunctor :
     shapes-polynomial-endofunctor 𝑃 → shapes-polynomial-endofunctor 𝑄
-  shapes-natural-transformation-polynomial-endofunctor a =
-    pr1 (type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α (a , id))
+  shapes-cartesian-natural-transformation-polynomial-endofunctor =
+    shapes-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
+          𝑃 𝑄 α)
 
-  positions-natural-transformation-polynomial-endofunctor :
+  positions-cartesian-natural-transformation-polynomial-endofunctor :
     (a : shapes-polynomial-endofunctor 𝑃) →
     positions-polynomial-endofunctor 𝑄
-      ( shapes-natural-transformation-polynomial-endofunctor a) →
+      ( shapes-cartesian-natural-transformation-polynomial-endofunctor a) →
     positions-polynomial-endofunctor 𝑃 a
-  positions-natural-transformation-polynomial-endofunctor a =
-    pr2 (type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α (a , id))
+  positions-cartesian-natural-transformation-polynomial-endofunctor =
+    positions-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
+          𝑃 𝑄 α)
 
-  hom-natural-transformation-polynomial-endofunctor :
+  hom-cartesian-natural-transformation-polynomial-endofunctor :
     hom-polynomial-endofunctor 𝑃 𝑄
-  hom-natural-transformation-polynomial-endofunctor =
-    ( shapes-natural-transformation-polynomial-endofunctor ,
-      positions-natural-transformation-polynomial-endofunctor)
+  hom-cartesian-natural-transformation-polynomial-endofunctor =
+    hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
+          𝑃 𝑄 α)
 ```
 
-### Equivalence between cartesian morphisms and cartesian natural transformations
-
-```text
-module _
-  {l1 l2 l3 l4 : Level}
-  (𝑃 : polynomial-endofunctor l1 l2)
-  (𝑄 : polynomial-endofunctor l3 l4)
-  where
-
-  is-retraction-hom-natural-transformation-polynomial-endofunctor :
-    is-retraction
-      ( λ α → natural-transformation-hom-polynomial-endofunctor 𝑃 𝑄 α {l2})
-      ( hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄)
-  is-retraction-hom-natural-transformation-polynomial-endofunctor α = refl
-
-  is-section-type-hom-natural-transformation-polynomial-endofunctor :
-    (α : natural-transformation-polynomial-endofunctor l2 𝑃 𝑄)
-    (X : UU l2) →
-    type-natural-transformation-polynomial-endofunctor 𝑃 𝑄
-      ( natural-transformation-hom-polynomial-endofunctor 𝑃 𝑄
-        ( hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α)) ~
-    type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α {X}
-  is-section-type-hom-natural-transformation-polynomial-endofunctor
-    α X (a , x) =
-    inv
-      ( naturality-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α x
-        ( a , id))
-```
+> TODO
diff --git a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
index 107b903571..3d1e32d549 100644
--- a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
@@ -13,6 +13,7 @@ open import foundation.binary-homotopies
 open import foundation.cartesian-morphisms-arrows
 open import foundation.commuting-squares-of-homotopies
 open import foundation.commuting-squares-of-maps
+open import foundation.cones-over-cospan-diagrams
 open import foundation.contractible-types
 open import foundation.dependent-pair-types
 open import foundation.equality-dependent-function-types
@@ -173,6 +174,15 @@ module _
   hom-arrow-cartesian-natural-transformation-polynomial-endofunctor =
     hom-arrow-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α'
 
+  cone-cartesian-natural-transformation-polynomial-endofunctor :
+    {X Y : UU l5} (f : X → Y) →
+    cone
+      ( type-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α)
+      ( map-polynomial-endofunctor 𝑄 f)
+      ( type-polynomial-endofunctor 𝑃 X)
+  cone-cartesian-natural-transformation-polynomial-endofunctor =
+    cone-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α'
+
   cartesian-hom-arrow-cartesian-natural-transformation-polynomial-endofunctor :
     {X Y : UU l5} (f : X → Y) →
     cartesian-hom-arrow
diff --git a/src/trees/morphisms-polynomial-endofunctors.lagda.md b/src/trees/morphisms-polynomial-endofunctors.lagda.md
index 901315003f..c415810061 100644
--- a/src/trees/morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/morphisms-polynomial-endofunctors.lagda.md
@@ -308,7 +308,7 @@ module _
 
   compute-fiber-type-hom-polynomial-endofunctor :
     (c : 𝑄₀) (x : 𝑄₁ c → X) →
-    fiber (type-hom-polynomial-endofunctor 𝑃 𝑄 α {X = X}) (c , x) ≃
+    fiber (type-hom-polynomial-endofunctor 𝑃 𝑄 α) (c , x) ≃
     fiber-type-hom-polynomial-endofunctor c x
   compute-fiber-type-hom-polynomial-endofunctor c x =
     equivalence-reasoning
@@ -353,7 +353,7 @@ module _
                       ( ap (λ q → x ∘ tr 𝑄₁ q) (inv-inv p)))))
 ```
 
-### Equivalence between morphisms and natural transformations
+### Comparison betwen morphisms and natural transformations
 
 ```agda
 module _
@@ -382,29 +382,8 @@ module _
         ( a , id))
 ```
 
-Finally, we need to show the following equality:
+To show these notions are equivalent, we need to show the following equality:
 
 $$
-  α₁ (f ∘ x) (a , id) = ap (𝑄 f) (α₁ x (a , id)) ∙ α₁ f (a , x)
+  α₁ (f ∘ x) (a , id) = ap (𝑄 f) (α₁ x (a , id)) ∙ α₁ f (a , x).
 $$
-
-```text
-module _
-  {l : Level}
-  (𝑃 : polynomial-endofunctor l l)
-  (𝑄 : polynomial-endofunctor l l)
-  where
-
-  is-section-is-set-positions-hom-natural-transformation-polynomial-endofunctor :
-    is-section
-      ( λ α → natural-transformation-hom-polynomial-endofunctor 𝑃 𝑄 α {l})
-      ( hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄)
-  is-section-is-set-positions-hom-natural-transformation-polynomial-endofunctor α =
-    eq-htpy-natural-transformation-polynomial-endofunctor 𝑃 𝑄
-      ( natural-transformation-hom-polynomial-endofunctor 𝑃 𝑄
-        ( hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α))
-      ( α)
-      ( ( is-section-type-hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄
-          ( α)) ,
-        ?)
-```

From f462576c44ba2459f34fd75f0e48aa5ad1897848 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 15 Sep 2025 14:32:33 +0200
Subject: [PATCH 18/41] more pullback pasting properties!

---
 src/foundation-core/pullbacks.lagda.md | 384 ++++++++++++++++++++-----
 1 file changed, 306 insertions(+), 78 deletions(-)

diff --git a/src/foundation-core/pullbacks.lagda.md b/src/foundation-core/pullbacks.lagda.md
index 0170c208fc..60c3346407 100644
--- a/src/foundation-core/pullbacks.lagda.md
+++ b/src/foundation-core/pullbacks.lagda.md
@@ -232,8 +232,8 @@ module _
 
   abstract
     is-pullback-htpy :
-      {c : cone f g C} (c' : cone f' g' C)
-      (Hc : htpy-parallel-cone Hf Hg c c') →
+      {c : cone f g C} (c' : cone f' g' C) →
+      htpy-parallel-cone Hf Hg c c' →
       is-pullback f' g' c' → is-pullback f g c
     is-pullback-htpy {c} c' H pb-c' =
       is-equiv-left-map-triangle
@@ -246,8 +246,8 @@ module _
 
   abstract
     is-pullback-htpy' :
-      (c : cone f g C) {c' : cone f' g' C}
-      (Hc : htpy-parallel-cone Hf Hg c c') →
+      (c : cone f g C) {c' : cone f' g' C} →
+      htpy-parallel-cone Hf Hg c c' →
       is-pullback f g c → is-pullback f' g' c'
     is-pullback-htpy' c {c'} H =
       is-equiv-top-map-triangle
@@ -533,74 +533,74 @@ module _
   {l1 l2 l3 l4 l5 l6 : Level}
   {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} {Y : UU l5} {Z : UU l6}
   (i : X → Y) (j : Y → Z) (h : C → Z)
-  (c : cone j h B) (d : cone i (vertical-map-cone j h c) A)
+  (right : cone j h B) (left : cone i (vertical-map-cone j h right) A)
   where
 
   abstract
     is-pullback-rectangle-is-pullback-left-square :
-      is-pullback j h c → is-pullback i (vertical-map-cone j h c) d →
-      is-pullback (j ∘ i) h (pasting-horizontal-cone i j h c d)
+      is-pullback j h right → is-pullback i (vertical-map-cone j h right) left →
+      is-pullback (j ∘ i) h (pasting-horizontal-cone i j h right left)
     is-pullback-rectangle-is-pullback-left-square pb-c pb-d =
       is-pullback-is-fiberwise-equiv-map-fiber-vertical-map-cone (j ∘ i) h
-        ( pasting-horizontal-cone i j h c d)
+        ( pasting-horizontal-cone i j h right left)
         ( λ x →
           is-equiv-left-map-triangle
             ( map-fiber-vertical-map-cone
-              ( j ∘ i) h (pasting-horizontal-cone i j h c d) x)
-            ( map-fiber-vertical-map-cone j h c (i x))
-            ( map-fiber-vertical-map-cone i (vertical-map-cone j h c) d x)
+              ( j ∘ i) h (pasting-horizontal-cone i j h right left) x)
+            ( map-fiber-vertical-map-cone j h right (i x))
+            ( map-fiber-vertical-map-cone i (vertical-map-cone j h right) left x)
             ( preserves-pasting-horizontal-map-fiber-vertical-map-cone
               ( i)
               ( j)
               ( h)
-              ( c)
-              ( d)
+              ( right)
+              ( left)
               ( x))
             ( is-fiberwise-equiv-map-fiber-vertical-map-cone-is-pullback
               ( i)
-              ( vertical-map-cone j h c)
-              ( d)
+              ( vertical-map-cone j h right)
+              ( left)
               ( pb-d)
               ( x))
             ( is-fiberwise-equiv-map-fiber-vertical-map-cone-is-pullback
               ( j)
               ( h)
-              ( c)
+              ( right)
               ( pb-c)
               ( i x)))
 
   abstract
     is-pullback-left-square-is-pullback-rectangle :
-      is-pullback j h c →
-      is-pullback (j ∘ i) h (pasting-horizontal-cone i j h c d) →
-      is-pullback i (vertical-map-cone j h c) d
+      is-pullback j h right →
+      is-pullback (j ∘ i) h (pasting-horizontal-cone i j h right left) →
+      is-pullback i (vertical-map-cone j h right) left
     is-pullback-left-square-is-pullback-rectangle pb-c pb-rect =
       is-pullback-is-fiberwise-equiv-map-fiber-vertical-map-cone i
-        ( vertical-map-cone j h c)
-        ( d)
+        ( vertical-map-cone j h right)
+        ( left)
         ( λ x →
           is-equiv-top-map-triangle
             ( map-fiber-vertical-map-cone
-              ( j ∘ i) h (pasting-horizontal-cone i j h c d) x)
-            ( map-fiber-vertical-map-cone j h c (i x))
-            ( map-fiber-vertical-map-cone i (vertical-map-cone j h c) d x)
+              ( j ∘ i) h (pasting-horizontal-cone i j h right left) x)
+            ( map-fiber-vertical-map-cone j h right (i x))
+            ( map-fiber-vertical-map-cone i (vertical-map-cone j h right) left x)
             ( preserves-pasting-horizontal-map-fiber-vertical-map-cone
               ( i)
               ( j)
               ( h)
-              ( c)
-              ( d)
+              ( right)
+              ( left)
               ( x))
             ( is-fiberwise-equiv-map-fiber-vertical-map-cone-is-pullback
               ( j)
               ( h)
-              ( c)
+              ( right)
               ( pb-c)
               ( i x))
             ( is-fiberwise-equiv-map-fiber-vertical-map-cone-is-pullback
               ( j ∘ i)
               ( h)
-              ( pasting-horizontal-cone i j h c d)
+              ( pasting-horizontal-cone i j h right left)
               ( pb-rect)
               ( x)))
 ```
@@ -628,107 +628,115 @@ module _
   {l1 l2 l3 l4 l5 l6 : Level}
   {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} {Y : UU l5} {Z : UU l6}
   (f : C → Z) (g : Y → Z) (h : X → Y)
-  (c : cone f g B) (d : cone (horizontal-map-cone f g c) h A)
+  (lower : cone f g B)
+  (upper : cone (horizontal-map-cone f g lower) h A)
   where
 
   abstract
     is-pullback-top-square-is-pullback-rectangle :
-      is-pullback f g c →
-      is-pullback f (g ∘ h) (pasting-vertical-cone f g h c d) →
-      is-pullback (horizontal-map-cone f g c) h d
-    is-pullback-top-square-is-pullback-rectangle pb-c pb-dc =
+      is-pullback f g lower →
+      is-pullback f (g ∘ h) (pasting-vertical-cone f g h lower upper) →
+      is-pullback (horizontal-map-cone f g lower) h upper
+    is-pullback-top-square-is-pullback-rectangle pb-lower pb-rect =
       is-pullback-is-fiberwise-equiv-map-fiber-vertical-map-cone
-        ( horizontal-map-cone f g c)
+        ( horizontal-map-cone f g lower)
         ( h)
-        ( d)
+        ( upper)
         ( λ x →
           is-fiberwise-equiv-is-equiv-map-Σ
             ( λ t → fiber h (pr1 t))
-            ( map-fiber-vertical-map-cone f g c (vertical-map-cone f g c x))
+            ( map-fiber-vertical-map-cone f g
+              ( lower)
+              ( vertical-map-cone f g lower x))
             ( λ t →
               map-fiber-vertical-map-cone
-                ( horizontal-map-cone f g c)
+                ( horizontal-map-cone f g lower)
                 ( h)
-                ( d)
+                ( upper)
                 ( pr1 t))
             ( is-fiberwise-equiv-map-fiber-vertical-map-cone-is-pullback
               ( f)
               ( g)
-              ( c)
-              ( pb-c)
-              ( vertical-map-cone f g c x))
+              ( lower)
+              ( pb-lower)
+              ( vertical-map-cone f g lower x))
             ( is-equiv-top-is-equiv-bottom-square
               ( map-inv-compute-fiber-comp
-                ( vertical-map-cone f g c)
-                ( vertical-map-cone (horizontal-map-cone f g c) h d)
-                ( vertical-map-cone f g c x))
-              ( map-inv-compute-fiber-comp g h (f (vertical-map-cone f g c x)))
+                ( vertical-map-cone f g lower)
+                ( vertical-map-cone (horizontal-map-cone f g lower) h upper)
+                ( vertical-map-cone f g lower x))
+              ( map-inv-compute-fiber-comp g h
+                ( f (vertical-map-cone f g lower x)))
               ( map-Σ
                 ( λ t → fiber h (pr1 t))
-                ( map-fiber-vertical-map-cone f g c (vertical-map-cone f g c x))
+                ( map-fiber-vertical-map-cone f g
+                  ( lower)
+                  ( vertical-map-cone f g lower x))
                 ( λ t →
                   map-fiber-vertical-map-cone
-                    ( horizontal-map-cone f g c) h d (pr1 t)))
+                    ( horizontal-map-cone f g lower) h upper (pr1 t)))
               ( map-fiber-vertical-map-cone f
                 ( g ∘ h)
-                ( pasting-vertical-cone f g h c d)
-                ( vertical-map-cone f g c x))
-              ( preserves-pasting-vertical-map-fiber-vertical-map-cone f g h c d
-                ( vertical-map-cone f g c x))
+                ( pasting-vertical-cone f g h lower upper)
+                ( vertical-map-cone f g lower x))
+              ( preserves-pasting-vertical-map-fiber-vertical-map-cone f g h
+                ( lower)
+                ( upper)
+                ( vertical-map-cone f g lower x))
               ( is-equiv-map-inv-compute-fiber-comp
-                ( vertical-map-cone f g c)
-                ( vertical-map-cone (horizontal-map-cone f g c) h d)
-                ( vertical-map-cone f g c x))
+                ( vertical-map-cone f g lower)
+                ( vertical-map-cone (horizontal-map-cone f g lower) h upper)
+                ( vertical-map-cone f g lower x))
               ( is-equiv-map-inv-compute-fiber-comp g h
-                ( f (vertical-map-cone f g c x)))
+                ( f (vertical-map-cone f g lower x)))
               ( is-fiberwise-equiv-map-fiber-vertical-map-cone-is-pullback f
                 ( g ∘ h)
-                ( pasting-vertical-cone f g h c d)
-                ( pb-dc)
-                ( vertical-map-cone f g c x)))
+                ( pasting-vertical-cone f g h lower upper)
+                ( pb-rect)
+                ( vertical-map-cone f g lower x)))
             ( x , refl))
 
   abstract
     is-pullback-rectangle-is-pullback-top-square :
-      is-pullback f g c →
-      is-pullback (horizontal-map-cone f g c) h d →
-      is-pullback f (g ∘ h) (pasting-vertical-cone f g h c d)
-    is-pullback-rectangle-is-pullback-top-square pb-c pb-d =
+      is-pullback f g lower →
+      is-pullback (horizontal-map-cone f g lower) h upper →
+      is-pullback f (g ∘ h) (pasting-vertical-cone f g h lower upper)
+    is-pullback-rectangle-is-pullback-top-square pb-lower pb-upper =
       is-pullback-is-fiberwise-equiv-map-fiber-vertical-map-cone
         ( f)
         ( g ∘ h)
-        ( pasting-vertical-cone f g h c d)
+        ( pasting-vertical-cone f g h lower upper)
         ( λ x →
           is-equiv-bottom-is-equiv-top-square
             ( map-inv-compute-fiber-comp
-              ( vertical-map-cone f g c)
-              ( vertical-map-cone (horizontal-map-cone f g c) h d)
+              ( vertical-map-cone f g lower)
+              ( vertical-map-cone (horizontal-map-cone f g lower) h upper)
               ( x))
             ( map-inv-compute-fiber-comp g h (f x))
             ( map-Σ
               ( λ t → fiber h (pr1 t))
-              ( map-fiber-vertical-map-cone f g c x)
+              ( map-fiber-vertical-map-cone f g lower x)
               ( λ t →
                 map-fiber-vertical-map-cone
-                  ( horizontal-map-cone f g c)
+                  ( horizontal-map-cone f g lower)
                   ( h)
-                  ( d)
+                  ( upper)
                   ( pr1 t)))
             ( map-fiber-vertical-map-cone
               ( f)
               ( g ∘ h)
-              ( pasting-vertical-cone f g h c d)
+              ( pasting-vertical-cone f g h lower upper)
               ( x))
             ( preserves-pasting-vertical-map-fiber-vertical-map-cone
               ( f)
               ( g)
               ( h)
-              ( c)
-              ( d)
+              ( lower)
+              ( upper)
               ( x))
             ( is-equiv-map-inv-compute-fiber-comp
-              ( vertical-map-cone f g c)
-              ( vertical-map-cone (horizontal-map-cone f g c) h d)
+              ( vertical-map-cone f g lower)
+              ( vertical-map-cone (horizontal-map-cone f g lower) h upper)
               ( x))
             ( is-equiv-map-inv-compute-fiber-comp g h (f x))
             ( is-equiv-map-Σ
@@ -736,18 +744,238 @@ module _
               ( is-fiberwise-equiv-map-fiber-vertical-map-cone-is-pullback
                 ( f)
                 ( g)
-                ( c)
-                ( pb-c)
+                ( lower)
+                ( pb-lower)
                 ( x))
               ( λ t →
                 is-fiberwise-equiv-map-fiber-vertical-map-cone-is-pullback
-                  ( horizontal-map-cone f g c)
+                  ( horizontal-map-cone f g lower)
                   ( h)
-                  ( d)
-                  ( pb-d)
+                  ( upper)
+                  ( pb-upper)
                   ( pr1 t))))
 ```
 
+### Triangles of pullback squares
+
+Given a triangle of squares where the back right square is a pullback,
+
+```text
+          •
+        ∧ | \
+       /  |  \
+      /   |   ∨
+    • --------> •
+    |     |     |
+    |     ∨     |
+    |     •     |
+    |   ∧   \   |
+    |  /     \  |
+    ∨ /       ∨ ∨
+    • --------> •
+```
+
+then the back left square is a pullback if and only if the front square is.
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} {Y : UU l5} {Z : UU l6}
+  (i : X → Y) (j : Y → Z) (h : C → Z) (k : X → Z)
+  (back-right : cone j h B)
+  (back-left : cone i (vertical-map-cone j h back-right) A)
+  (front : cone k h A)
+  (K : j ∘ i ~ k)
+  (H :
+    htpy-parallel-cone
+      ( K)
+      ( refl-htpy' h)
+      ( pasting-horizontal-cone i j h back-right back-left)
+      ( front))
+  where
+
+  abstract
+    is-pullback-left-square-horizontal-triangle :
+      is-pullback j h back-right →
+      is-pullback i (vertical-map-cone j h back-right) back-left →
+      is-pullback k h front
+    is-pullback-left-square-horizontal-triangle pb-back-right pb-back-left =
+      is-pullback-htpy'
+        ( K)
+        ( refl-htpy' h)
+        ( pasting-horizontal-cone i j h back-right back-left)
+        ( H)
+        ( is-pullback-rectangle-is-pullback-left-square i j h
+          ( back-right)
+          ( back-left)
+          ( pb-back-right)
+          ( pb-back-left))
+
+  abstract
+    is-pullback-front-square-horizontal-triangle :
+      is-pullback j h back-right →
+      is-pullback k h front →
+      is-pullback i (vertical-map-cone j h back-right) back-left
+    is-pullback-front-square-horizontal-triangle pb-back-right pb-front =
+      is-pullback-left-square-is-pullback-rectangle i j h back-right back-left
+        ( pb-back-right)
+        ( is-pullback-htpy K (refl-htpy' h) front H pb-front)
+
+module _
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} {Y : UU l5} {Z : UU l6}
+  (i : X → Y) (j : Y → Z) (h : C → Z) (k : X → Z)
+  (back-right : cone j h B)
+  (back-left : cone i (vertical-map-cone j h back-right) A)
+  (front : cone k h A)
+  (K : k ~ j ∘ i)
+  (H :
+    htpy-parallel-cone
+      ( K)
+      ( refl-htpy' h)
+      ( front)
+      ( pasting-horizontal-cone i j h back-right back-left))
+  where
+
+  abstract
+    is-pullback-left-square-horizontal-triangle' :
+      is-pullback j h back-right →
+      is-pullback i (vertical-map-cone j h back-right) back-left →
+      is-pullback k h front
+    is-pullback-left-square-horizontal-triangle' pb-back-right pb-back-left =
+      is-pullback-htpy
+        ( K)
+        ( refl-htpy' h)
+        ( pasting-horizontal-cone i j h back-right back-left)
+        ( H)
+        ( is-pullback-rectangle-is-pullback-left-square i j h
+          ( back-right)
+          ( back-left)
+          ( pb-back-right)
+          ( pb-back-left))
+
+  abstract
+    is-pullback-front-square-horizontal-triangle' :
+      is-pullback j h back-right →
+      is-pullback k h front →
+      is-pullback i (vertical-map-cone j h back-right) back-left
+    is-pullback-front-square-horizontal-triangle' pb-back-right pb-front =
+      is-pullback-left-square-is-pullback-rectangle i j h
+        ( back-right)
+        ( back-left)
+        ( pb-back-right)
+        ( is-pullback-htpy' K (refl-htpy' h) front H pb-front)
+```
+
+We can also glue along horizontal maps.
+
+Given a triangle of squares where the right square is a pullback,
+
+```text
+         αX
+  • -----------> •
+   \  ⟍           \   ⟍
+    \    ↘          \    ↘
+     \     • ------- \ --> •
+      \   / ⌟         \   /
+       ∨ ∨             ∨ ∨
+        • ------------> •
+```
+
+then the left square is a pullback if and only if the top square is.
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} {Y : UU l5} {Z : UU l6}
+  (f : C → Z) (g : Y → Z) (h : X → Y) (k : X → Z)
+  (right : cone f g B) (top : cone (horizontal-map-cone f g right) h A)
+  (left : cone f k A)
+  (K : g ∘ h ~ k)
+  (H :
+    htpy-parallel-cone
+      ( refl-htpy' f)
+      ( K)
+      ( pasting-vertical-cone f g h right top)
+      ( left))
+  where
+
+  abstract
+    is-pullback-left-square-vertical-triangle :
+      is-pullback f g right →
+      is-pullback (horizontal-map-cone f g right) h top →
+      is-pullback f k left
+    is-pullback-left-square-vertical-triangle pb-right pb-top =
+      is-pullback-htpy'
+        ( refl-htpy' f)
+        ( K)
+        ( pasting-vertical-cone f g h right top)
+        ( H)
+        ( is-pullback-rectangle-is-pullback-top-square f g h right top
+          ( pb-right)
+          ( pb-top))
+
+  abstract
+    is-pullback-top-square-vertical-triangle :
+      is-pullback f g right →
+      is-pullback f k left →
+      is-pullback (horizontal-map-cone f g right) h top
+    is-pullback-top-square-vertical-triangle pb-right pb-left =
+      is-pullback-top-square-is-pullback-rectangle f g h right top
+        ( pb-right)
+        ( is-pullback-htpy
+          ( refl-htpy' f)
+          ( K)
+          ( left)
+          ( H)
+          ( pb-left))
+
+module _
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} {Y : UU l5} {Z : UU l6}
+  (f : C → Z) (g : Y → Z) (h : X → Y) (k : X → Z)
+  (right : cone f g B) (top : cone (horizontal-map-cone f g right) h A)
+  (left : cone f k A)
+  (K : k ~ g ∘ h)
+  (H :
+    htpy-parallel-cone
+      ( refl-htpy' f)
+      ( K)
+      ( left)
+      ( pasting-vertical-cone f g h right top))
+  where
+
+  abstract
+    is-pullback-left-square-vertical-triangle' :
+      is-pullback f g right →
+      is-pullback (horizontal-map-cone f g right) h top →
+      is-pullback f k left
+    is-pullback-left-square-vertical-triangle' pb-right pb-top =
+      is-pullback-htpy
+        ( refl-htpy' f)
+        ( K)
+        ( pasting-vertical-cone f g h right top)
+        ( H)
+        ( is-pullback-rectangle-is-pullback-top-square f g h right top
+          ( pb-right)
+          ( pb-top))
+
+  abstract
+    is-pullback-top-square-vertical-triangle' :
+      is-pullback f g right →
+      is-pullback f k left →
+      is-pullback (horizontal-map-cone f g right) h top
+    is-pullback-top-square-vertical-triangle' pb-right pb-left =
+      is-pullback-top-square-is-pullback-rectangle f g h right top
+        ( pb-right)
+        ( is-pullback-htpy'
+          ( refl-htpy' f)
+          ( K)
+          ( left)
+          ( H)
+          ( pb-left))
+```
+
 ### Pullbacks are associative
 
 Consider two cospans with a shared vertex `B`:

From 5b4c5bfeb616e7b27e1671ba8301bf77abde7545 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 15 Sep 2025 14:54:51 +0200
Subject: [PATCH 19/41] edits

---
 src/foundation-core/sections.lagda.md         |  2 +-
 .../cartesian-morphisms-arrows.lagda.md       | 21 +++++++++++++++++++
 ...morphisms-polynomial-endofunctors.lagda.md |  2 +-
 ...ormations-polynomial-endofunctors.lagda.md |  8 +++++++
 4 files changed, 31 insertions(+), 2 deletions(-)

diff --git a/src/foundation-core/sections.lagda.md b/src/foundation-core/sections.lagda.md
index fbf0044e3d..fdcf282625 100644
--- a/src/foundation-core/sections.lagda.md
+++ b/src/foundation-core/sections.lagda.md
@@ -97,7 +97,7 @@ module _
   is-section-map-section-comp :
     is-section (g ∘ h) map-section-comp
   is-section-map-section-comp =
-    ( g ·l (is-section-map-section h t ·r map-section g s)) ∙h
+    ( g ·l is-section-map-section h t ·r map-section g s) ∙h
     ( is-section-map-section g s)
 
   section-comp : section (g ∘ h)
diff --git a/src/foundation/cartesian-morphisms-arrows.lagda.md b/src/foundation/cartesian-morphisms-arrows.lagda.md
index 58ea9ebfbb..e47f256ea1 100644
--- a/src/foundation/cartesian-morphisms-arrows.lagda.md
+++ b/src/foundation/cartesian-morphisms-arrows.lagda.md
@@ -253,6 +253,27 @@ module _
       is-cartesian-transpose-cartesian-hom-arrow)
 ```
 
+### If the transpose is cartesian then so is the original
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4}
+  (f : A → B) (g : X → Y) (α : hom-arrow f g)
+  where
+
+  is-cartesian-is-cartesian-transpose-cartesian-hom-arrow :
+    is-cartesian-hom-arrow
+      ( map-domain-hom-arrow f g α)
+      ( map-codomain-hom-arrow f g α)
+      ( transpose-hom-arrow f g α) →
+    is-cartesian-hom-arrow f g α
+  is-cartesian-is-cartesian-transpose-cartesian-hom-arrow =
+    is-pullback-swap-cone'
+      ( map-codomain-hom-arrow f g α)
+      ( g)
+      ( cone-hom-arrow f g α)
+```
+
 ### If the target of a cartesian morphism is an equivalence then so is the source
 
 ```agda
diff --git a/src/trees/morphisms-polynomial-endofunctors.lagda.md b/src/trees/morphisms-polynomial-endofunctors.lagda.md
index c415810061..6ea8706dfc 100644
--- a/src/trees/morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/morphisms-polynomial-endofunctors.lagda.md
@@ -353,7 +353,7 @@ module _
                       ( ap (λ q → x ∘ tr 𝑄₁ q) (inv-inv p)))))
 ```
 
-### Comparison betwen morphisms and natural transformations
+### Comparison between morphisms and natural transformations
 
 ```agda
 module _
diff --git a/src/trees/natural-transformations-polynomial-endofunctors.lagda.md b/src/trees/natural-transformations-polynomial-endofunctors.lagda.md
index 72b9e168f2..0830fcab48 100644
--- a/src/trees/natural-transformations-polynomial-endofunctors.lagda.md
+++ b/src/trees/natural-transformations-polynomial-endofunctors.lagda.md
@@ -130,6 +130,14 @@ module _
   hom-arrow-natural-transformation-polynomial-endofunctor =
     ( α₀ , α₀ , α₁ f)
 
+  transpose-hom-arrow-natural-transformation-polynomial-endofunctor :
+    hom-arrow (α₀ {X}) (α₀ {Y})
+  transpose-hom-arrow-natural-transformation-polynomial-endofunctor =
+    transpose-hom-arrow
+      ( map-polynomial-endofunctor 𝑃 f)
+      ( map-polynomial-endofunctor 𝑄 f)
+      ( hom-arrow-natural-transformation-polynomial-endofunctor)
+
   cone-natural-transformation-polynomial-endofunctor :
     cone α₀ (map-polynomial-endofunctor 𝑄 f) (type-polynomial-endofunctor 𝑃 X)
   cone-natural-transformation-polynomial-endofunctor =

From 6d60ca976286af23b463907f928b3dd024487d2f Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 15 Sep 2025 14:55:55 +0200
Subject: [PATCH 20/41] add reference for equivalence between cartesian
 morphisms and cartesian natural transformations

---
 references.bib                                     | 14 ++++++++++++++
 ...sian-morphisms-polynomial-endofunctors.lagda.md | 13 +++++++++++--
 2 files changed, 25 insertions(+), 2 deletions(-)

diff --git a/references.bib b/references.bib
index da0e6f65ee..2bda1c4a2a 100644
--- a/references.bib
+++ b/references.bib
@@ -520,6 +520,20 @@ @article{GK12
   issn          = {1469-8064},
 }
 
+@article{HGK22,
+  title         = {∞-Operads as Analytic Monads},
+  author        = {Gepner, David and Haugseng, Rune and Kock, Joachim},
+  year          = 2021,
+  month         = {04},
+  journal       = {International Mathematics Research Notices},
+  volume        = 2022,
+  number        = 16,
+  pages         = {12516--12624},
+  doi           = {10.1093/imrn/rnaa332},
+  issn          = {1073-7928},
+  eprint        = {https://academic.oup.com/imrn/article-pdf/2022/16/12516/45280093/rnaa332.pdf},
+}
+
 @book{Johnstone02,
   title         = {Sketches of an Elephant a Topos Theory Compendium},
   author        = {Johnstone, Peter T},
diff --git a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
index a33bd8b4df..61d0a18ff4 100644
--- a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
@@ -344,7 +344,7 @@ module _
                 ( inv-tr (λ c → positions-polynomial-endofunctor 𝑄 c → X) p x))
 ```
 
-### Cartesian morphisms are cartesian natural transformations
+### The associated natural transformation of a cartesian morphism is cartesian
 
 ```agda
 module _
@@ -380,7 +380,7 @@ module _
             ( is-equiv-id))
 ```
 
-### Cartesian natural transformations define cartesian morphisms
+### The associated morphism of a cartesian natural transformation is cartesian
 
 ```agda
 module _
@@ -419,3 +419,12 @@ module _
 ```
 
 > TODO
+
+### Equivalence between cartesian morphisms and cartesian natural transformations
+
+This is proved in a slightly different setting as part of Lemma 2.3.3 in
+{{#cite GHK22}}.
+
+## References
+
+{{#bibliography}}

From 42e5612761a607fe4403f3593fd4bfd201c84cf0 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 15 Sep 2025 23:27:32 +0200
Subject: [PATCH 21/41] uniqueness of lifts cartesian morphisms of arrows

---
 .../cartesian-morphisms-arrows.lagda.md       | 117 +++++++++++++++++-
 src/foundation/homotopies.lagda.md            |   2 +-
 ...ormations-polynomial-endofunctors.lagda.md |   2 +-
 3 files changed, 113 insertions(+), 8 deletions(-)

diff --git a/src/foundation/cartesian-morphisms-arrows.lagda.md b/src/foundation/cartesian-morphisms-arrows.lagda.md
index e47f256ea1..809a00b83c 100644
--- a/src/foundation/cartesian-morphisms-arrows.lagda.md
+++ b/src/foundation/cartesian-morphisms-arrows.lagda.md
@@ -15,9 +15,14 @@ open import foundation.coproducts-pullbacks
 open import foundation.dependent-pair-types
 open import foundation.dependent-products-pullbacks
 open import foundation.dependent-sums-pullbacks
+open import foundation.transport-along-identifications
 open import foundation.diagonal-maps-cartesian-products-of-types
 open import foundation.equivalences
+open import foundation.function-extensionality
 open import foundation.fibers-of-maps
+open import foundation.contractible-types
+open import foundation.torsorial-type-families
+open import foundation.homotopies
 open import foundation.function-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.functoriality-coproduct-types
@@ -718,7 +723,7 @@ module _
       ( is-cartesian-coproduct-cartesian-hom-arrow))
 ```
 
-### Cartesian morphisms of arrows are preserved under exponentiation
+### Cartesian morphisms of arrows are preserved under postcomposition exponentiation
 
 ```agda
 module _
@@ -777,9 +782,9 @@ is cartesian if and only if either of the folded morphisms
              Δ                             Δ
 ```
 
-is.
+is cartesian.
 
-It remains to formalize the right-hand version.
+> It remains to formalize the right-hand version.
 
 ```agda
 module _
@@ -906,7 +911,7 @@ if `α` is.
 
 **Proof.** The unique existence of `γ` and the homotopy follows from the
 pullback property of `β`. The rest is a reiteration of the 3-for-2 property of
-cartesian morphisms.
+cartesian morphisms. ∎
 
 We begin by constructing the commuting triangle of morphisms of arrows:
 
@@ -1043,6 +1048,103 @@ module _
       ( inv-coherence-triangle-hom-arrow-lift-map-codomain-cartesian-hom-arrow)
 ```
 
+This lift is unique.
+
+<!-- The following uniqueness proof also constructs an explicit lift, making the entire formalization above almost entirely useless -->
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (f : A → A') (g : B → B') (h : C → C')
+  (β : cartesian-hom-arrow g h)
+  (α : hom-arrow f h)
+  (i : A' → B')
+  (H :
+    coherence-triangle-maps'
+      ( map-codomain-hom-arrow f h α)
+      ( map-codomain-cartesian-hom-arrow g h β)
+      ( i))
+  where
+
+  abstract
+    uniqueness-lift-map-codomain-cartesian-hom-arrow :
+      is-torsorial
+        ( λ (j : A → B) →
+          Σ ( coherence-hom-arrow f g j i)
+            ( λ J →
+              Σ ( coherence-triangle-maps'
+                  ( map-domain-hom-arrow f h α)
+                  ( map-domain-cartesian-hom-arrow g h β)
+                  ( j))
+                ( λ K →
+                  coherence-htpy-hom-arrow f h
+                    ( comp-hom-arrow f g h
+                      ( hom-arrow-cartesian-hom-arrow g h β)
+                      ( j , i , J))
+                    ( α)
+                    ( K)
+                    ( H))))
+    uniqueness-lift-map-codomain-cartesian-hom-arrow =
+      is-contr-equiv _
+        ( equiv-tot
+          ( λ j →
+            equiv-Σ _
+              ( equiv-inv-htpy (i ∘ f) (g ∘ j))
+              ( λ J →
+                equiv-tot
+                  ( λ K →
+                    equivalence-reasoning
+                    ( ( map-codomain-cartesian-hom-arrow g h β ·l J) ∙h
+                      ( coh-cartesian-hom-arrow g h β ·r j) ∙h
+                      ( h ·l K) ~
+                      ( H ·r f) ∙h
+                      ( coh-hom-arrow f h α))
+                    ≃ ( ( map-codomain-cartesian-hom-arrow g h β ·l J) ∙h
+                        ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h
+                          ( h ·l K)) ~
+                        ( H ·r f) ∙h
+                        ( coh-hom-arrow f h α))
+                      by
+                        equiv-tr
+                          ( λ Q → Q ~ ( H ·r f) ∙h ( coh-hom-arrow f h α))
+                          ( eq-htpy
+                            ( assoc-htpy
+                              ( map-codomain-cartesian-hom-arrow g h β ·l J)
+                              ( coh-cartesian-hom-arrow g h β ·r j)
+                              ( h ·l K)))
+                    ≃ ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K) ~
+                        ( inv-htpy
+                          ( map-codomain-cartesian-hom-arrow g h β ·l J)) ∙h
+                        ( ( H ·r f) ∙h (coh-hom-arrow f h α)))
+                      by
+                        equiv-left-transpose-htpy-concat
+                          ( map-codomain-cartesian-hom-arrow g h β ·l J)
+                          ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K))
+                          ( ( H ·r f) ∙h (coh-hom-arrow f h α))
+                    ≃ ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K) ~
+                        ( ( map-codomain-cartesian-hom-arrow g h β) ·l
+                          ( inv-htpy J)) ∙h
+                        ( ( H ·r f) ∙h (coh-hom-arrow f h α)))
+                      by
+                        equiv-tr
+                          ( λ Q →
+                            (coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K) ~
+                            Q ∙h ((H ·r f) ∙h (coh-hom-arrow f h α)))
+                          ( inv
+                            ( eq-htpy
+                              ( left-whisker-inv-htpy
+                                ( map-codomain-cartesian-hom-arrow g h β)
+                                ( J))))))))
+        ( uniqueness-universal-property-pullback
+          ( map-codomain-cartesian-hom-arrow g h β)
+          ( h)
+          ( cone-cartesian-hom-arrow g h β)
+          ( universal-property-cartesian-hom-arrow g h β)
+          ( A)
+          ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow f g h β α i H))
+```
+
 Now, if `α` was cartesian to begin with, the lift is also.
 
 ```agda
@@ -1070,8 +1172,11 @@ module _
           ( H))
     is-cartesian-cartesian-hom-arrow-lift-map-codomain-cartesian-hom-arrow =
       is-cartesian-top-cartesian-hom-arrow-triangle' f g h
-        ( hom-arrow-lift-map-codomain-cartesian-hom-arrow
-          ( f) g h β (hom-arrow-cartesian-hom-arrow f h α) i H)
+        ( hom-arrow-lift-map-codomain-cartesian-hom-arrow f g h
+          ( β)
+          ( hom-arrow-cartesian-hom-arrow f h α)
+          ( i)
+          ( H))
         ( α)
         ( β)
         ( inv-coherence-triangle-hom-arrow-lift-map-codomain-cartesian-hom-arrow
diff --git a/src/foundation/homotopies.lagda.md b/src/foundation/homotopies.lagda.md
index 54f138533e..88a2ba84ea 100644
--- a/src/foundation/homotopies.lagda.md
+++ b/src/foundation/homotopies.lagda.md
@@ -225,7 +225,7 @@ module _
     is-equiv-map-Π-is-fiberwise-equiv
       ( λ x → is-equiv-left-transpose-eq-concat (H x) (K x) (L x))
 
-  equiv-left-transpose-htpy-concat : ((H ∙h K) ~ L) ≃ (K ~ ((inv-htpy H) ∙h L))
+  equiv-left-transpose-htpy-concat : (H ∙h K ~ L) ≃ (K ~ (inv-htpy H ∙h L))
   pr1 equiv-left-transpose-htpy-concat = left-transpose-htpy-concat H K L
   pr2 equiv-left-transpose-htpy-concat = is-equiv-left-transpose-htpy-concat
 
diff --git a/src/trees/natural-transformations-polynomial-endofunctors.lagda.md b/src/trees/natural-transformations-polynomial-endofunctors.lagda.md
index 0830fcab48..f2ddd62628 100644
--- a/src/trees/natural-transformations-polynomial-endofunctors.lagda.md
+++ b/src/trees/natural-transformations-polynomial-endofunctors.lagda.md
@@ -82,7 +82,7 @@ coherence-natural-transformation-polynomial-endofunctor {l = l} 𝑃 𝑄 α₀
     ( α₀)
     ( map-polynomial-endofunctor 𝑃 f)
     ( map-polynomial-endofunctor 𝑄 f)
-  ( α₀)
+    ( α₀)
 
 natural-transformation-polynomial-endofunctor :
   {l1 l2 l3 l4 : Level} (l : Level) →

From fb0dab724fcc6dafe98f09a14e09b414019be1f5 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Mon, 15 Sep 2025 23:27:38 +0200
Subject: [PATCH 22/41] a comment

---
 src/trees/morphisms-polynomial-endofunctors.lagda.md | 8 ++++++--
 1 file changed, 6 insertions(+), 2 deletions(-)

diff --git a/src/trees/morphisms-polynomial-endofunctors.lagda.md b/src/trees/morphisms-polynomial-endofunctors.lagda.md
index 6ea8706dfc..56c62529c1 100644
--- a/src/trees/morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/morphisms-polynomial-endofunctors.lagda.md
@@ -382,8 +382,12 @@ module _
         ( a , id))
 ```
 
-To show these notions are equivalent, we need to show the following equality:
+**Comment.** If we were to show these notions are equivalent, we would have
+needed natural transformations to satisfy the following equality:
 
 $$
-  α₁ (f ∘ x) (a , id) = ap (𝑄 f) (α₁ x (a , id)) ∙ α₁ f (a , x).
+  α₁ (f ∘ x) (a , id) = ap (𝑄 f) (α₁ x (a , id)) ∙ α₁ f (a , x),
 $$
+
+which is an unfolding of the condition that the naturality square of a composite
+map is given by pasting of squares.

From 55e0caaad4eab418b51ca4e255cc13435d34de9c Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Tue, 16 Sep 2025 00:11:06 +0200
Subject: [PATCH 23/41] lifts of morphisms of arrows

---
 src/foundation.lagda.md                       |  1 +
 .../lifts-morphisms-arrows.lagda.md           | 93 +++++++++++++++++++
 2 files changed, 94 insertions(+)
 create mode 100644 src/foundation/lifts-morphisms-arrows.lagda.md

diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md
index f0f52a7493..f23ecc41cf 100644
--- a/src/foundation.lagda.md
+++ b/src/foundation.lagda.md
@@ -290,6 +290,7 @@ open import foundation.large-locale-of-subtypes public
 open import foundation.law-of-excluded-middle public
 open import foundation.lawveres-fixed-point-theorem public
 open import foundation.lesser-limited-principle-of-omniscience public
+open import foundation.lifts-morphisms-arrows public
 open import foundation.lifts-types public
 open import foundation.limited-principle-of-omniscience public
 open import foundation.locale-of-propositions public
diff --git a/src/foundation/lifts-morphisms-arrows.lagda.md b/src/foundation/lifts-morphisms-arrows.lagda.md
new file mode 100644
index 0000000000..18feb7e14c
--- /dev/null
+++ b/src/foundation/lifts-morphisms-arrows.lagda.md
@@ -0,0 +1,93 @@
+# Lifts of morphisms of arrows
+
+```agda
+module foundation.lifts-morphisms-arrows where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.commuting-triangles-of-morphisms-arrows
+open import foundation.dependent-pair-types
+open import foundation.morphisms-arrows
+open import foundation.homotopies-morphisms-arrows
+open import foundation.universe-levels
+open import foundation.commuting-triangles-of-maps
+```
+
+</details>
+
+## Idea
+
+Given two [morphism of arrows](foundation.morphisms-arrows.md) `α : f ⇒ h` and
+`β : g ⇒ h`, then a
+{{#concept "lift" Disambiguation="morphisms of arrows" Agda=lift-hom-arrow}} of
+`α` along `β` is a morphism of arrows `γ : f ⇒ g` such that the triangle
+`β ∘ γ ~ α` commutes.
+
+## Definition
+
+```agda
+lift-hom-arrow :
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (f : A → A') (g : B → B') (h : C → C')
+  (α : hom-arrow f h) (β : hom-arrow g h) →
+  UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6)
+lift-hom-arrow f g h α β =
+  Σ (hom-arrow f g) (coherence-triangle-hom-arrow f g h α β)
+
+module _
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (f : A → A') (g : B → B') (h : C → C')
+  (α : hom-arrow f h) (β : hom-arrow g h)
+  (γ : lift-hom-arrow f g h α β)
+  where
+
+  hom-arrow-lift-hom-arrow : hom-arrow f g
+  hom-arrow-lift-hom-arrow = pr1 γ
+
+  coherence-triangle-lift-hom-arrow :
+    coherence-triangle-hom-arrow f g h α β hom-arrow-lift-hom-arrow
+  coherence-triangle-lift-hom-arrow = pr2 γ
+
+  map-domain-lift-hom-arrow : A → B
+  map-domain-lift-hom-arrow =
+    map-domain-hom-arrow f g hom-arrow-lift-hom-arrow
+
+  map-codomain-lift-hom-arrow : A' → B'
+  map-codomain-lift-hom-arrow =
+    map-codomain-hom-arrow f g hom-arrow-lift-hom-arrow
+
+  coh-hom-arrow-lift-hom-arrow :
+    coherence-hom-arrow f g
+      map-domain-lift-hom-arrow
+      map-codomain-lift-hom-arrow
+  coh-hom-arrow-lift-hom-arrow =
+    coh-hom-arrow f g hom-arrow-lift-hom-arrow
+
+  coh-domain-lift-hom-arrow :
+    coherence-triangle-maps
+      ( map-domain-hom-arrow f h α)
+      ( map-domain-hom-arrow g h β)
+      ( map-domain-lift-hom-arrow)
+  coh-domain-lift-hom-arrow =
+    pr1 coherence-triangle-lift-hom-arrow
+
+  coh-codomain-lift-hom-arrow :
+    coherence-triangle-maps
+      ( map-codomain-hom-arrow f h α)
+      ( map-codomain-hom-arrow g h β)
+      ( map-codomain-lift-hom-arrow)
+  coh-codomain-lift-hom-arrow =
+    pr1 (pr2 coherence-triangle-lift-hom-arrow)
+
+  coh-lift-hom-arrow :
+    coherence-htpy-hom-arrow f h α
+      ( comp-hom-arrow f g h β hom-arrow-lift-hom-arrow)
+      ( coh-domain-lift-hom-arrow)
+      ( coh-codomain-lift-hom-arrow)
+  coh-lift-hom-arrow =
+    pr2 (pr2 coherence-triangle-lift-hom-arrow)
+```

From 881c9119c7318b09272e58e024590434bf52826d Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Tue, 16 Sep 2025 01:33:39 +0200
Subject: [PATCH 24/41] wip universal property cartesian morphisms

---
 references.bib                                |  28 +-
 src/foundation-core/pullbacks.lagda.md        |  12 +-
 src/foundation.lagda.md                       |   1 +
 .../cartesian-morphisms-arrows.lagda.md       | 325 +---------
 .../lifts-morphisms-arrows.lagda.md           |   4 +-
 src/foundation/subuniverses.lagda.md          |   3 +-
 ...operty-cartesian-morphisms-arrows.lagda.md | 611 ++++++++++++++++++
 ...ormations-polynomial-endofunctors.lagda.md | 120 ++++
 8 files changed, 767 insertions(+), 337 deletions(-)
 create mode 100644 src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md

diff --git a/references.bib b/references.bib
index 2bda1c4a2a..a906360dac 100644
--- a/references.bib
+++ b/references.bib
@@ -506,6 +506,20 @@ @online{GGMS24
   keywords      = {Computer Science - Logic in Computer Science,Mathematics - Logic},
 }
 
+@article{GHK22,
+  title         = {∞-Operads as Analytic Monads},
+  author        = {Gepner, David and Haugseng, Rune and Kock, Joachim},
+  year          = 2021,
+  month         = {04},
+  journal       = {International Mathematics Research Notices},
+  volume        = 2022,
+  number        = 16,
+  pages         = {12516--12624},
+  doi           = {10.1093/imrn/rnaa332},
+  issn          = {1073-7928},
+  eprint        = {https://academic.oup.com/imrn/article-pdf/2022/16/12516/45280093/rnaa332.pdf},
+}
+
 @article{GK12,
   title         = {Polynomial functors and polynomial monads},
   author        = {GAMBINO, NICOLA and KOCK, JOACHIM},
@@ -520,20 +534,6 @@ @article{GK12
   issn          = {1469-8064},
 }
 
-@article{HGK22,
-  title         = {∞-Operads as Analytic Monads},
-  author        = {Gepner, David and Haugseng, Rune and Kock, Joachim},
-  year          = 2021,
-  month         = {04},
-  journal       = {International Mathematics Research Notices},
-  volume        = 2022,
-  number        = 16,
-  pages         = {12516--12624},
-  doi           = {10.1093/imrn/rnaa332},
-  issn          = {1073-7928},
-  eprint        = {https://academic.oup.com/imrn/article-pdf/2022/16/12516/45280093/rnaa332.pdf},
-}
-
 @book{Johnstone02,
   title         = {Sketches of an Elephant a Topos Theory Compendium},
   author        = {Johnstone, Peter T},
diff --git a/src/foundation-core/pullbacks.lagda.md b/src/foundation-core/pullbacks.lagda.md
index 60c3346407..78f9f1379c 100644
--- a/src/foundation-core/pullbacks.lagda.md
+++ b/src/foundation-core/pullbacks.lagda.md
@@ -548,7 +548,11 @@ module _
             ( map-fiber-vertical-map-cone
               ( j ∘ i) h (pasting-horizontal-cone i j h right left) x)
             ( map-fiber-vertical-map-cone j h right (i x))
-            ( map-fiber-vertical-map-cone i (vertical-map-cone j h right) left x)
+            ( map-fiber-vertical-map-cone
+              ( i)
+              ( vertical-map-cone j h right)
+              ( left)
+              ( x))
             ( preserves-pasting-horizontal-map-fiber-vertical-map-cone
               ( i)
               ( j)
@@ -583,7 +587,11 @@ module _
             ( map-fiber-vertical-map-cone
               ( j ∘ i) h (pasting-horizontal-cone i j h right left) x)
             ( map-fiber-vertical-map-cone j h right (i x))
-            ( map-fiber-vertical-map-cone i (vertical-map-cone j h right) left x)
+            ( map-fiber-vertical-map-cone
+              ( i)
+              ( vertical-map-cone j h right)
+              ( left)
+              ( x))
             ( preserves-pasting-horizontal-map-fiber-vertical-map-cone
               ( i)
               ( j)
diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md
index f23ecc41cf..6da8339e48 100644
--- a/src/foundation.lagda.md
+++ b/src/foundation.lagda.md
@@ -489,6 +489,7 @@ open import foundation.univalence public
 open import foundation.univalence-implies-function-extensionality public
 open import foundation.univalent-type-families public
 open import foundation.universal-property-booleans public
+open import foundation.universal-property-cartesian-morphisms-arrows public
 open import foundation.universal-property-cartesian-product-types public
 open import foundation.universal-property-contractible-types public
 open import foundation.universal-property-coproduct-types public
diff --git a/src/foundation/cartesian-morphisms-arrows.lagda.md b/src/foundation/cartesian-morphisms-arrows.lagda.md
index 809a00b83c..288eadafba 100644
--- a/src/foundation/cartesian-morphisms-arrows.lagda.md
+++ b/src/foundation/cartesian-morphisms-arrows.lagda.md
@@ -11,23 +11,21 @@ open import foundation.action-on-identifications-functions
 open import foundation.commuting-triangles-of-maps
 open import foundation.commuting-triangles-of-morphisms-arrows
 open import foundation.cones-over-cospan-diagrams
+open import foundation.contractible-types
 open import foundation.coproducts-pullbacks
 open import foundation.dependent-pair-types
 open import foundation.dependent-products-pullbacks
 open import foundation.dependent-sums-pullbacks
-open import foundation.transport-along-identifications
 open import foundation.diagonal-maps-cartesian-products-of-types
 open import foundation.equivalences
-open import foundation.function-extensionality
 open import foundation.fibers-of-maps
-open import foundation.contractible-types
-open import foundation.torsorial-type-families
-open import foundation.homotopies
+open import foundation.function-extensionality
 open import foundation.function-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.functoriality-coproduct-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.functoriality-fibers-of-maps
+open import foundation.homotopies
 open import foundation.homotopies-morphisms-arrows
 open import foundation.identity-types
 open import foundation.morphisms-arrows
@@ -36,12 +34,13 @@ open import foundation.postcomposition-pullbacks
 open import foundation.products-pullbacks
 open import foundation.pullbacks
 open import foundation.standard-pullbacks
+open import foundation.torsorial-type-families
+open import foundation.transport-along-identifications
 open import foundation.unit-type
 open import foundation.universe-levels
 open import foundation.whiskering-homotopies-composition
 
 open import foundation-core.commuting-squares-of-maps
-open import foundation-core.homotopies
 open import foundation-core.propositions
 open import foundation-core.universal-property-pullbacks
 ```
@@ -135,12 +134,12 @@ module _
   cone-cartesian-hom-arrow =
     cone-hom-arrow f g hom-arrow-cartesian-hom-arrow
 
-  universal-property-cartesian-hom-arrow :
+  up-pullback-cartesian-hom-arrow :
     universal-property-pullback
       ( map-codomain-cartesian-hom-arrow)
       ( g)
       ( cone-cartesian-hom-arrow)
-  universal-property-cartesian-hom-arrow =
+  up-pullback-cartesian-hom-arrow =
     universal-property-pullback-is-pullback
       ( map-codomain-cartesian-hom-arrow)
       ( g)
@@ -880,316 +879,6 @@ module _
       ( is-cartesian-cartesian-hom-arrow f g α)
 ```
 
-### Lifting cartesian morphisms along lifts of the codomain
-
-Suppose given a cospan diagram of arrows
-
-```text
-    A ------> C <------ B
-    |         |       ⌞ |
-  f |    α    h    β    | g
-    ∨         ∨         ∨
-    A' -----> C' <----- B'
-```
-
-where `β` is cartesian. Moreover, suppose we have a map `i : A' → B'` from the
-codomain of the source of `α` to the codomain of the source of `β` such that the
-triangle
-
-```text
-         i
-     A' ---> B'
-      \     /
-       \   /
-        ∨ ∨
-         C'
-```
-
-commutes. Then there is a unique morphism of arrows `γ : f → g` with a homotopy
-`β ~ α ∘ γ` extending the triangle, and this morphism is cartesian if and only
-if `α` is.
-
-**Proof.** The unique existence of `γ` and the homotopy follows from the
-pullback property of `β`. The rest is a reiteration of the 3-for-2 property of
-cartesian morphisms. ∎
-
-We begin by constructing the commuting triangle of morphisms of arrows:
-
-```agda
-module _
-  {l1 l2 l3 l4 l5 l6 : Level}
-  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
-  (f : A → A') (g : B → B') (h : C → C')
-  (β : cartesian-hom-arrow g h)
-  (α : hom-arrow f h)
-  (i : A' → B')
-  (H :
-    coherence-triangle-maps'
-      ( map-codomain-hom-arrow f h α)
-      ( map-codomain-cartesian-hom-arrow g h β)
-      ( i))
-  where
-
-  cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow :
-    cone (map-codomain-cartesian-hom-arrow g h β) h A
-  cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow =
-    ( i ∘ f , map-domain-hom-arrow f h α , H ·r f ∙h coh-hom-arrow f h α)
-
-  map-domain-hom-arrow-lift-map-codomain-cartesian-hom-arrow : A → B
-  map-domain-hom-arrow-lift-map-codomain-cartesian-hom-arrow =
-    gap-is-pullback
-      ( map-codomain-cartesian-hom-arrow g h β)
-      ( h)
-      ( cone-cartesian-hom-arrow g h β)
-      ( is-cartesian-cartesian-hom-arrow g h β)
-      ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow)
-
-  hom-arrow-lift-map-codomain-cartesian-hom-arrow : hom-arrow f g
-  pr1 hom-arrow-lift-map-codomain-cartesian-hom-arrow =
-    map-domain-hom-arrow-lift-map-codomain-cartesian-hom-arrow
-  pr1 (pr2 hom-arrow-lift-map-codomain-cartesian-hom-arrow) = i
-  pr2 (pr2 hom-arrow-lift-map-codomain-cartesian-hom-arrow) =
-    inv-htpy
-      ( htpy-vertical-map-gap-is-pullback
-        ( map-codomain-cartesian-hom-arrow g h β)
-        ( h)
-        ( cone-cartesian-hom-arrow g h β)
-        ( is-cartesian-cartesian-hom-arrow g h β)
-        ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow))
-
-  abstract
-    inv-coherence-triangle-hom-arrow-lift-map-codomain-cartesian-hom-arrow :
-      coherence-triangle-hom-arrow' f g h
-        ( α)
-        ( hom-arrow-cartesian-hom-arrow g h β)
-        ( hom-arrow-lift-map-codomain-cartesian-hom-arrow)
-    inv-coherence-triangle-hom-arrow-lift-map-codomain-cartesian-hom-arrow =
-      ( htpy-horizontal-map-gap-is-pullback
-          ( map-codomain-cartesian-hom-arrow g h β)
-          ( h)
-          ( cone-cartesian-hom-arrow g h β)
-          ( is-cartesian-cartesian-hom-arrow g h β)
-          ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow)) ,
-      ( H) ,
-      ( ( ap-concat-htpy'
-          ( ( h) ·l
-            ( htpy-horizontal-map-gap-is-pullback
-              ( map-codomain-cartesian-hom-arrow g h β)
-              ( h)
-              ( cone-cartesian-hom-arrow g h β)
-              ( pr2 β)
-              ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow)))
-          ( ap-concat-htpy'
-            ( coh-cartesian-hom-arrow g h β ·r
-              map-domain-hom-arrow-lift-map-codomain-cartesian-hom-arrow)
-            ( left-whisker-inv-htpy
-              ( map-codomain-cartesian-hom-arrow g h β)
-              ( htpy-vertical-map-gap-is-pullback
-                ( map-codomain-cartesian-hom-arrow g h β)
-                ( h)
-                ( cone-cartesian-hom-arrow g h β)
-                ( pr2 β)
-                ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow))))) ∙h
-        ( assoc-htpy
-          ( inv-htpy
-            ( ( map-codomain-cartesian-hom-arrow g h β) ·l
-              ( htpy-vertical-map-gap-is-pullback
-                ( map-codomain-cartesian-hom-arrow g h β)
-                ( h)
-                ( cone-cartesian-hom-arrow g h β)
-                ( pr2 β)
-                ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow))))
-          ( coh-cartesian-hom-arrow g h β ·r
-            map-domain-hom-arrow-lift-map-codomain-cartesian-hom-arrow)
-          ( ( h) ·l
-            ( htpy-horizontal-map-gap-is-pullback
-              ( map-codomain-cartesian-hom-arrow g h β)
-              ( h)
-              ( cone-cartesian-hom-arrow g h β)
-              ( pr2 β)
-              ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow)))) ∙h
-        ( inv-htpy-left-transpose-htpy-concat
-          ( ( map-codomain-cartesian-hom-arrow g h β) ·l
-            ( htpy-vertical-map-gap-is-pullback
-              ( map-codomain-cartesian-hom-arrow g h β)
-              ( h)
-              ( cone-cartesian-hom-arrow g h β)
-              ( pr2 β)
-              ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow)))
-          ( H ·r f ∙h coh-hom-arrow f h α)
-          ( ( coh-cartesian-hom-arrow g h β ·r
-              map-domain-hom-arrow-lift-map-codomain-cartesian-hom-arrow) ∙h
-            ( h) ·l
-            ( htpy-horizontal-map-gap-is-pullback
-              ( map-codomain-cartesian-hom-arrow g h β)
-              ( h)
-              ( cone-cartesian-hom-arrow g h β)
-              ( is-cartesian-cartesian-hom-arrow g h β)
-              ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow)))
-          ( inv-htpy
-            ( coh-htpy-cone-gap-is-pullback
-              ( map-codomain-cartesian-hom-arrow g h β)
-              ( h)
-              ( cone-cartesian-hom-arrow g h β)
-              ( is-cartesian-cartesian-hom-arrow g h β)
-              ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow)))))
-
-  coherence-triangle-hom-arrow-lift-map-codomain-cartesian-hom-arrow :
-    coherence-triangle-hom-arrow f g h
-      ( α)
-      ( hom-arrow-cartesian-hom-arrow g h β)
-      ( hom-arrow-lift-map-codomain-cartesian-hom-arrow)
-  coherence-triangle-hom-arrow-lift-map-codomain-cartesian-hom-arrow =
-    inv-htpy-hom-arrow f h
-      ( comp-hom-arrow f g h
-        ( hom-arrow-cartesian-hom-arrow g h β)
-        ( hom-arrow-lift-map-codomain-cartesian-hom-arrow))
-      ( α)
-      ( inv-coherence-triangle-hom-arrow-lift-map-codomain-cartesian-hom-arrow)
-```
-
-This lift is unique.
-
-<!-- The following uniqueness proof also constructs an explicit lift, making the entire formalization above almost entirely useless -->
-
-```agda
-module _
-  {l1 l2 l3 l4 l5 l6 : Level}
-  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
-  (f : A → A') (g : B → B') (h : C → C')
-  (β : cartesian-hom-arrow g h)
-  (α : hom-arrow f h)
-  (i : A' → B')
-  (H :
-    coherence-triangle-maps'
-      ( map-codomain-hom-arrow f h α)
-      ( map-codomain-cartesian-hom-arrow g h β)
-      ( i))
-  where
-
-  abstract
-    uniqueness-lift-map-codomain-cartesian-hom-arrow :
-      is-torsorial
-        ( λ (j : A → B) →
-          Σ ( coherence-hom-arrow f g j i)
-            ( λ J →
-              Σ ( coherence-triangle-maps'
-                  ( map-domain-hom-arrow f h α)
-                  ( map-domain-cartesian-hom-arrow g h β)
-                  ( j))
-                ( λ K →
-                  coherence-htpy-hom-arrow f h
-                    ( comp-hom-arrow f g h
-                      ( hom-arrow-cartesian-hom-arrow g h β)
-                      ( j , i , J))
-                    ( α)
-                    ( K)
-                    ( H))))
-    uniqueness-lift-map-codomain-cartesian-hom-arrow =
-      is-contr-equiv _
-        ( equiv-tot
-          ( λ j →
-            equiv-Σ _
-              ( equiv-inv-htpy (i ∘ f) (g ∘ j))
-              ( λ J →
-                equiv-tot
-                  ( λ K →
-                    equivalence-reasoning
-                    ( ( map-codomain-cartesian-hom-arrow g h β ·l J) ∙h
-                      ( coh-cartesian-hom-arrow g h β ·r j) ∙h
-                      ( h ·l K) ~
-                      ( H ·r f) ∙h
-                      ( coh-hom-arrow f h α))
-                    ≃ ( ( map-codomain-cartesian-hom-arrow g h β ·l J) ∙h
-                        ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h
-                          ( h ·l K)) ~
-                        ( H ·r f) ∙h
-                        ( coh-hom-arrow f h α))
-                      by
-                        equiv-tr
-                          ( λ Q → Q ~ ( H ·r f) ∙h ( coh-hom-arrow f h α))
-                          ( eq-htpy
-                            ( assoc-htpy
-                              ( map-codomain-cartesian-hom-arrow g h β ·l J)
-                              ( coh-cartesian-hom-arrow g h β ·r j)
-                              ( h ·l K)))
-                    ≃ ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K) ~
-                        ( inv-htpy
-                          ( map-codomain-cartesian-hom-arrow g h β ·l J)) ∙h
-                        ( ( H ·r f) ∙h (coh-hom-arrow f h α)))
-                      by
-                        equiv-left-transpose-htpy-concat
-                          ( map-codomain-cartesian-hom-arrow g h β ·l J)
-                          ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K))
-                          ( ( H ·r f) ∙h (coh-hom-arrow f h α))
-                    ≃ ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K) ~
-                        ( ( map-codomain-cartesian-hom-arrow g h β) ·l
-                          ( inv-htpy J)) ∙h
-                        ( ( H ·r f) ∙h (coh-hom-arrow f h α)))
-                      by
-                        equiv-tr
-                          ( λ Q →
-                            (coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K) ~
-                            Q ∙h ((H ·r f) ∙h (coh-hom-arrow f h α)))
-                          ( inv
-                            ( eq-htpy
-                              ( left-whisker-inv-htpy
-                                ( map-codomain-cartesian-hom-arrow g h β)
-                                ( J))))))))
-        ( uniqueness-universal-property-pullback
-          ( map-codomain-cartesian-hom-arrow g h β)
-          ( h)
-          ( cone-cartesian-hom-arrow g h β)
-          ( universal-property-cartesian-hom-arrow g h β)
-          ( A)
-          ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow f g h β α i H))
-```
-
-Now, if `α` was cartesian to begin with, the lift is also.
-
-```agda
-module _
-  {l1 l2 l3 l4 l5 l6 : Level}
-  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
-  (f : A → A') (g : B → B') (h : C → C')
-  (β : cartesian-hom-arrow g h)
-  (α : cartesian-hom-arrow f h)
-  (i : A' → B')
-  (H :
-    coherence-triangle-maps'
-      ( map-codomain-cartesian-hom-arrow f h α)
-      ( map-codomain-cartesian-hom-arrow g h β)
-      ( i))
-  where
-
-  abstract
-    is-cartesian-cartesian-hom-arrow-lift-map-codomain-cartesian-hom-arrow :
-      is-cartesian-hom-arrow f g
-        ( hom-arrow-lift-map-codomain-cartesian-hom-arrow f g h
-          ( β)
-          ( hom-arrow-cartesian-hom-arrow f h α)
-          ( i)
-          ( H))
-    is-cartesian-cartesian-hom-arrow-lift-map-codomain-cartesian-hom-arrow =
-      is-cartesian-top-cartesian-hom-arrow-triangle' f g h
-        ( hom-arrow-lift-map-codomain-cartesian-hom-arrow f g h
-          ( β)
-          ( hom-arrow-cartesian-hom-arrow f h α)
-          ( i)
-          ( H))
-        ( α)
-        ( β)
-        ( inv-coherence-triangle-hom-arrow-lift-map-codomain-cartesian-hom-arrow
-          ( f) g h β (hom-arrow-cartesian-hom-arrow f h α) i H)
-
-  cartesian-hom-arrow-lift-map-codomain-cartesian-hom-arrow :
-    cartesian-hom-arrow f g
-  cartesian-hom-arrow-lift-map-codomain-cartesian-hom-arrow =
-    ( hom-arrow-lift-map-codomain-cartesian-hom-arrow
-      ( f) g h β (hom-arrow-cartesian-hom-arrow f h α) i H) ,
-    ( is-cartesian-cartesian-hom-arrow-lift-map-codomain-cartesian-hom-arrow)
-```
-
 ## See also
 
 - [Cocartesian morphisms of arrows](synthetic-homotopy-theory.cocartesian-morphisms-arrows.md)
diff --git a/src/foundation/lifts-morphisms-arrows.lagda.md b/src/foundation/lifts-morphisms-arrows.lagda.md
index 18feb7e14c..aa2f4f318a 100644
--- a/src/foundation/lifts-morphisms-arrows.lagda.md
+++ b/src/foundation/lifts-morphisms-arrows.lagda.md
@@ -7,12 +7,12 @@ module foundation.lifts-morphisms-arrows where
 <details><summary>Imports</summary>
 
 ```agda
+open import foundation.commuting-triangles-of-maps
 open import foundation.commuting-triangles-of-morphisms-arrows
 open import foundation.dependent-pair-types
-open import foundation.morphisms-arrows
 open import foundation.homotopies-morphisms-arrows
+open import foundation.morphisms-arrows
 open import foundation.universe-levels
-open import foundation.commuting-triangles-of-maps
 ```
 
 </details>
diff --git a/src/foundation/subuniverses.lagda.md b/src/foundation/subuniverses.lagda.md
index 2a19ec8bfd..b655d7cafc 100644
--- a/src/foundation/subuniverses.lagda.md
+++ b/src/foundation/subuniverses.lagda.md
@@ -90,7 +90,8 @@ module _
     Σ (type-subuniverse P) (λ Y → inclusion-subuniverse P Y ≃ X)
 
   is-proof-irrelevant-is-essentially-in-subuniverse :
-    {l3 : Level} (X : UU l3) → is-proof-irrelevant (is-essentially-in-subuniverse X)
+    {l3 : Level} (X : UU l3) →
+    is-proof-irrelevant (is-essentially-in-subuniverse X)
   is-proof-irrelevant-is-essentially-in-subuniverse X ((X' , p) , e) =
     is-torsorial-Eq-subtype
       ( is-contr-equiv'
diff --git a/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md b/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
new file mode 100644
index 0000000000..d379555370
--- /dev/null
+++ b/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
@@ -0,0 +1,611 @@
+# The universal property of cartesian morphisms of arrows
+
+```agda
+module foundation.universal-property-cartesian-morphisms-arrows where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.action-on-identifications-functions
+open import foundation.cartesian-morphisms-arrows
+open import foundation.commuting-triangles-of-maps
+open import foundation.commuting-triangles-of-morphisms-arrows
+open import foundation.cones-over-cospan-diagrams
+open import foundation.contractible-types
+open import foundation.coproducts-pullbacks
+open import foundation.dependent-pair-types
+open import foundation.dependent-products-pullbacks
+open import foundation.dependent-sums-pullbacks
+open import foundation.diagonal-maps-cartesian-products-of-types
+open import foundation.equivalences
+open import foundation.fibers-of-maps
+open import foundation.function-extensionality
+open import foundation.function-types
+open import foundation.functoriality-cartesian-product-types
+open import foundation.functoriality-coproduct-types
+open import foundation.functoriality-dependent-pair-types
+open import foundation.functoriality-fibers-of-maps
+open import foundation.homotopies
+open import foundation.homotopies-morphisms-arrows
+open import foundation.identity-types
+open import foundation.lifts-morphisms-arrows
+open import foundation.morphisms-arrows
+open import foundation.postcomposition-functions
+open import foundation.postcomposition-pullbacks
+open import foundation.products-pullbacks
+open import foundation.pullbacks
+open import foundation.standard-pullbacks
+open import foundation.torsorial-type-families
+open import foundation.transport-along-identifications
+open import foundation.unit-type
+open import foundation.universe-levels
+open import foundation.whiskering-homotopies-composition
+
+open import foundation-core.commuting-squares-of-maps
+open import foundation-core.propositions
+open import foundation-core.universal-property-pullbacks
+
+open import orthogonal-factorization-systems.lifts-maps
+```
+
+</details>
+
+## Idea
+
+A [morphism of arrows](foundation.morphisms-arrows.md) `β : g ⇒ h`,
+
+```text
+         β₀
+    B ------> C
+    |         |
+  g |    H    | h
+    ∨         ∨
+    B' -----> C',
+         β₁
+```
+
+is said to satisfy the
+{{#concept "universal property" Disambiguation="cartesian morphisms of arrows"}}
+of cartesian morphisms of arrows if the natural map that assigns to every
+extension diagram of the form
+
+```text
+          B
+        ∧ | \
+       /  |  \
+      /   g   ∨
+    A --------> C
+    |     |     |
+    |     ∨  H  |
+  f |     B'    | h
+    |   ∧   \β₁ |
+    |  /     \  |
+    ∨ /       ∨ ∨
+    A' -------> C'
+```
+
+the underlying data
+
+```text
+    A --------> C
+    |           |
+    |        H  |
+  f |     B'    | h
+    |   ∧   \β₁ |
+    | i/     \  |
+    ∨ /       ∨ ∨
+    A' -------> C'
+```
+
+is an equivalence.
+
+## Definitions
+
+```agda
+lift-codomain-lift-hom-arrow :
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (f : A → A') (g : B → B') (h : C → C')
+  (α : hom-arrow f h) (β : hom-arrow g h) →
+  lift-hom-arrow f g h α β →
+  lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α)
+lift-codomain-lift-hom-arrow f g h α β γ =
+  ( map-codomain-lift-hom-arrow f g h α β γ ,
+    coh-codomain-lift-hom-arrow f g h α β γ)
+```
+
+### The universal property of cartesian morphisms of arrows
+
+```agda
+universal-property-cartesian-hom-arrow-Level :
+  (l1 l2 : Level) {l3 l4 l5 l6 : Level}
+  {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (g : B → B') (h : C → C') (β : hom-arrow g h) →
+  UU (lsuc l1 ⊔ lsuc l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6)
+universal-property-cartesian-hom-arrow-Level l1 l2 g h β =
+  {A : UU l1} {A' : UU l2} (f : A → A') (α : hom-arrow f h) →
+  is-equiv (lift-codomain-lift-hom-arrow f g h α β)
+
+universal-property-cartesian-hom-arrow :
+  {l3 l4 l5 l6 : Level}
+  {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (g : B → B') (h : C → C') (β : hom-arrow g h) →
+  UUω
+universal-property-cartesian-hom-arrow g h β =
+  {l1 l2 : Level} → universal-property-cartesian-hom-arrow-Level l1 l2 g h β
+```
+
+## Properties
+
+### Computing the fibers of `lift-codomain-lift-hom-arrow`
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (f : A → A') (g : B → B') (h : C → C')
+  (α : hom-arrow f h) (β : hom-arrow g h)
+  where
+
+  family-fiber-lift-codomain-lift-hom-arrow :
+    lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α) →
+    (A → B) → UU (l1 ⊔ l4 ⊔ l5 ⊔ l6)
+  family-fiber-lift-codomain-lift-hom-arrow (i , I) j =
+    Σ (coherence-hom-arrow f g j i)
+      ( λ H →
+        Σ ( coherence-triangle-maps
+            ( map-domain-hom-arrow f h α)
+            ( map-domain-hom-arrow g h β)
+            ( j))
+          ( λ J →
+            coherence-htpy-hom-arrow f h α
+              ( comp-hom-arrow f g h β (j , i , H))
+              ( J)
+              ( I)))
+
+  fiber-lift-codomain-lift-hom-arrow :
+    lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α) →
+    UU (l1 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6)
+  fiber-lift-codomain-lift-hom-arrow i =
+    Σ (A → B) (family-fiber-lift-codomain-lift-hom-arrow i)
+```
+
+> The explicit equivalence remains to be written out.
+
+### Uniqueness of lifts of cartesian morphisms
+
+Given a cartesian morphism of arrows, then lifts are unique.
+
+<!-- The following equivalence uses function extensionality in many places it does not need to -->
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (f : A → A') (g : B → B') (h : C → C')
+  (α : hom-arrow f h) (β : hom-arrow g h)
+  ((i , I) : lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α))
+  where
+
+  abstract
+    equiv-htpy-cone-family-fiber-lift-codomain-lift-hom-arrow :
+      (j : A → B) →
+      family-fiber-lift-codomain-lift-hom-arrow f g h α β (i , I) j ≃
+      htpy-cone (map-codomain-hom-arrow g h β) h
+        ( g ∘ j ,
+          map-domain-hom-arrow g h β ∘ j ,
+          coh-hom-arrow g h β ·r j)
+        ( i ∘ f ,
+          map-domain-hom-arrow f h α ,
+          inv-htpy I ·r f ∙h coh-hom-arrow f h α)
+    equiv-htpy-cone-family-fiber-lift-codomain-lift-hom-arrow j =
+      equiv-Σ _
+        ( equiv-inv-htpy (i ∘ f) (g ∘ j))
+        ( λ H →
+          equiv-Σ _
+            ( equiv-inv-htpy (pr1 α) (pr1 β ∘ j))
+            ( λ J →
+              equivalence-reasoning
+              ( ( coh-hom-arrow f h α ∙h h ·l J) ~
+                ( I ·r f) ∙h
+                ( ( map-codomain-hom-arrow g h β ·l H) ∙h
+                  ( coh-hom-arrow g h β ·r j)))
+              ≃ ( ( I ·r f) ∙h
+                  ( ( map-codomain-hom-arrow g h β ·l H) ∙h
+                    ( coh-hom-arrow g h β ·r j)) ~
+                  ( coh-hom-arrow f h α ∙h h ·l J))
+                by equiv-inv-htpy _ _
+              ≃ ( ( map-codomain-hom-arrow g h β ·l H) ∙h
+                  ( coh-hom-arrow g h β ·r j) ~
+                  ( inv-htpy I ·r f) ∙h (coh-hom-arrow f h α ∙h h ·l J))
+                by
+                  equiv-left-transpose-htpy-concat
+                    ( I ·r f)
+                    ( ( map-codomain-hom-arrow g h β ·l H) ∙h
+                      ( coh-hom-arrow g h β ·r j))
+                    ( coh-hom-arrow f h α ∙h h ·l J)
+              ≃ ( ( coh-hom-arrow g h β ·r j) ~
+                  ( inv-htpy (map-codomain-hom-arrow g h β ·l H)) ∙h
+                  ( inv-htpy I ·r f ∙h (coh-hom-arrow f h α ∙h h ·l J)))
+                by
+                  equiv-left-transpose-htpy-concat
+                    ( map-codomain-hom-arrow g h β ·l H)
+                    ( coh-hom-arrow g h β ·r j)
+                    ( inv-htpy I ·r f ∙h (coh-hom-arrow f h α ∙h h ·l J))
+              ≃ ( ( coh-hom-arrow g h β ·r j) ~
+                  ( map-codomain-hom-arrow g h β ·l (inv-htpy H)) ∙h
+                  ( inv-htpy I ·r f ∙h (coh-hom-arrow f h α ∙h h ·l J)))
+                by
+                  equiv-tr
+                    ( λ Q →
+                      coh-hom-arrow g h β ·r j ~
+                      Q ∙h (inv-htpy I ·r f ∙h (coh-hom-arrow f h α ∙h h ·l J)))
+                    ( inv
+                      ( eq-htpy
+                        ( left-whisker-inv-htpy
+                          ( map-codomain-hom-arrow g h β) H)))
+              ≃ ( ( coh-hom-arrow g h β ·r j) ~
+                  ( ( map-codomain-hom-arrow g h β ·l (inv-htpy H)) ∙h
+                    ( inv-htpy I ·r f ∙h (coh-hom-arrow f h α))) ∙h
+                  ( h ·l J))
+                by
+                  equiv-tr
+                    ( coh-hom-arrow g h β ·r j ~_)
+                    ( ( ap
+                        ( map-codomain-hom-arrow g h β ·l (inv-htpy H) ∙h_)
+                        ( eq-htpy
+                          ( inv-htpy-assoc-htpy
+                            ( inv-htpy I ·r f)
+                            ( coh-hom-arrow f h α)
+                            ( h ·l J)))) ∙
+                      ( eq-htpy
+                        ( inv-htpy-assoc-htpy
+                          ( map-codomain-hom-arrow g h β ·l (inv-htpy H))
+                          ( inv-htpy I ·r f ∙h (coh-hom-arrow f h α))
+                          ( h ·l J))))
+              ≃ ( ( coh-hom-arrow g h β ·r j) ∙h (inv-htpy (h ·l J)) ~
+                  ( map-codomain-hom-arrow g h β ·l (inv-htpy H)) ∙h
+                  ( (inv-htpy I ·r f) ∙h (coh-hom-arrow f h α)))
+                by
+                  equiv-right-transpose-htpy-concat'
+                    ( coh-hom-arrow g h β ·r j)
+                    ( ( map-codomain-hom-arrow g h β ·l (inv-htpy H)) ∙h
+                      ( inv-htpy I ·r f ∙h (coh-hom-arrow f h α)))
+                    ( h ·l J)
+              ≃ ( ( coh-hom-arrow g h β ·r j) ∙h (h ·l inv-htpy J) ~
+                  ( map-codomain-hom-arrow g h β ·l inv-htpy H) ∙h
+                  ( ( inv-htpy I ·r f) ∙h (coh-hom-arrow f h α)))
+                by
+                  equiv-tr
+                    ( λ Q →
+                      ( coh-hom-arrow g h β ·r j) ∙h Q ~
+                      ( map-codomain-hom-arrow g h β ·l inv-htpy H) ∙h
+                      ( ( inv-htpy I ·r f) ∙h (coh-hom-arrow f h α)))
+                    ( inv (eq-htpy (left-whisker-inv-htpy h J)))))
+
+  abstract
+    uniqueness-lift-map-codomain-is-cartesian-hom-arrow :
+      is-cartesian-hom-arrow g h β →
+      is-torsorial (family-fiber-lift-codomain-lift-hom-arrow f g h α β (i , I))
+    uniqueness-lift-map-codomain-is-cartesian-hom-arrow H =
+      is-contr-equiv _
+        ( equiv-tot equiv-htpy-cone-family-fiber-lift-codomain-lift-hom-arrow)
+        ( uniqueness-universal-property-pullback
+          ( map-codomain-hom-arrow g h β)
+          ( h)
+          ( cone-hom-arrow g h β)
+          ( up-pullback-cartesian-hom-arrow g h (β , H))
+          ( A)
+          ( _))
+```
+
+<!-- Theis uniqueness proof also constructs an explicit lift, making the entire next section entirely useless -->
+
+> What follows is another computation of the same. Should be deleted.
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (f : A → A') (g : B → B') (h : C → C')
+  (β : cartesian-hom-arrow g h)
+  (α : hom-arrow f h)
+  (i : A' → B')
+  (H :
+    coherence-triangle-maps'
+      ( map-codomain-hom-arrow f h α)
+      ( map-codomain-cartesian-hom-arrow g h β)
+      ( i))
+  where
+
+  abstract
+    uniqueness-lift-map-codomain-cartesian-hom-arrow' :
+      is-torsorial
+        ( λ (j : A → B) →
+          Σ ( coherence-hom-arrow f g j i)
+            ( λ J →
+              Σ ( coherence-triangle-maps'
+                  ( map-domain-hom-arrow f h α)
+                  ( map-domain-cartesian-hom-arrow g h β)
+                  ( j))
+                ( λ K →
+                  coherence-htpy-hom-arrow f h
+                    ( comp-hom-arrow f g h
+                      ( hom-arrow-cartesian-hom-arrow g h β)
+                      ( j , i , J))
+                    ( α)
+                    ( K)
+                    ( H))))
+    uniqueness-lift-map-codomain-cartesian-hom-arrow' =
+      is-contr-equiv ((Σ (A → B) λ j → htpy-cone (map-codomain-cartesian-hom-arrow g h β) h ((λ z → g (j z)) , (λ z → pr1 (pr1 β) (j z)) , (λ x → pr2 (pr2 (pr1 β)) (j x))) ((λ z → i (f z)) , pr1 α , H ·r f ∙h coh-hom-arrow f h α)))
+        ( equiv-tot
+          ( λ j →
+            equiv-Σ _
+              ( equiv-inv-htpy (i ∘ f) (g ∘ j))
+              ( λ J →
+                equiv-tot
+                  ( λ K →
+                    equivalence-reasoning
+                    ( ( map-codomain-cartesian-hom-arrow g h β ·l J) ∙h
+                      ( coh-cartesian-hom-arrow g h β ·r j) ∙h
+                      ( h ·l K) ~
+                      ( H ·r f) ∙h
+                      ( coh-hom-arrow f h α))
+                    ≃ ( ( map-codomain-cartesian-hom-arrow g h β ·l J) ∙h
+                        ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h
+                          ( h ·l K)) ~
+                        ( H ·r f) ∙h
+                        ( coh-hom-arrow f h α))
+                      by
+                        equiv-tr
+                          ( λ Q → Q ~ ( H ·r f) ∙h ( coh-hom-arrow f h α))
+                          ( eq-htpy
+                            ( assoc-htpy
+                              ( map-codomain-cartesian-hom-arrow g h β ·l J)
+                              ( coh-cartesian-hom-arrow g h β ·r j)
+                              ( h ·l K)))
+                    ≃ ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K) ~
+                        ( inv-htpy
+                          ( map-codomain-cartesian-hom-arrow g h β ·l J)) ∙h
+                        ( ( H ·r f) ∙h (coh-hom-arrow f h α)))
+                      by
+                        equiv-left-transpose-htpy-concat
+                          ( map-codomain-cartesian-hom-arrow g h β ·l J)
+                          ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K))
+                          ( ( H ·r f) ∙h (coh-hom-arrow f h α))
+                    ≃ ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K) ~
+                        ( ( map-codomain-cartesian-hom-arrow g h β) ·l
+                          ( inv-htpy J)) ∙h
+                        ( ( H ·r f) ∙h (coh-hom-arrow f h α)))
+                      by
+                        equiv-tr
+                          ( λ Q →
+                            (coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K) ~
+                            Q ∙h ((H ·r f) ∙h (coh-hom-arrow f h α)))
+                          ( inv
+                            ( eq-htpy
+                              ( left-whisker-inv-htpy
+                                ( map-codomain-cartesian-hom-arrow g h β)
+                                ( J))))))))
+        ( uniqueness-universal-property-pullback
+          ( map-codomain-cartesian-hom-arrow g h β)
+          ( h)
+          ( cone-cartesian-hom-arrow g h β)
+          ( up-pullback-cartesian-hom-arrow g h β)
+          ( A)
+          ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow f g h β α i H))
+```
+
+### Lifting cartesian morphisms along lifts of the codomain
+
+Suppose given a cospan diagram of arrows
+
+```text
+    A ------> C <------ B
+    |         |       ⌞ |
+  f |    α    h    β    | g
+    ∨         ∨         ∨
+    A' -----> C' <----- B'
+```
+
+where `β` is cartesian. Moreover, suppose we have a map `i : A' → B'` from the
+codomain of the source of `α` to the codomain of the source of `β` such that the
+triangle
+
+```text
+         i
+     A' ---> B'
+      \     /
+       \   /
+        ∨ ∨
+         C'
+```
+
+commutes. Then there is a unique morphism of arrows `γ : f → g` with a homotopy
+`β ~ α ∘ γ` extending the triangle, and this morphism is cartesian if and only
+if `α` is.
+
+**Proof.** The unique existence of `γ` and the homotopy follows from the
+pullback property of `β`. The rest is a reiteration of the 3-for-2 property of
+cartesian morphisms. ∎
+
+We begin by constructing the commuting triangle of morphisms of arrows:
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (f : A → A') (g : B → B') (h : C → C')
+  (β : cartesian-hom-arrow g h)
+  (α : hom-arrow f h)
+  (i : A' → B')
+  (H :
+    coherence-triangle-maps'
+      ( map-codomain-hom-arrow f h α)
+      ( map-codomain-cartesian-hom-arrow g h β)
+      ( i))
+  where
+
+  cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow :
+    cone (map-codomain-cartesian-hom-arrow g h β) h A
+  cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow =
+    ( i ∘ f , map-domain-hom-arrow f h α , H ·r f ∙h coh-hom-arrow f h α)
+
+  map-domain-hom-arrow-lift-map-codomain-cartesian-hom-arrow : A → B
+  map-domain-hom-arrow-lift-map-codomain-cartesian-hom-arrow =
+    gap-is-pullback
+      ( map-codomain-cartesian-hom-arrow g h β)
+      ( h)
+      ( cone-cartesian-hom-arrow g h β)
+      ( is-cartesian-cartesian-hom-arrow g h β)
+      ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow)
+
+  hom-arrow-lift-map-codomain-cartesian-hom-arrow : hom-arrow f g
+  pr1 hom-arrow-lift-map-codomain-cartesian-hom-arrow =
+    map-domain-hom-arrow-lift-map-codomain-cartesian-hom-arrow
+  pr1 (pr2 hom-arrow-lift-map-codomain-cartesian-hom-arrow) = i
+  pr2 (pr2 hom-arrow-lift-map-codomain-cartesian-hom-arrow) =
+    inv-htpy
+      ( htpy-vertical-map-gap-is-pullback
+        ( map-codomain-cartesian-hom-arrow g h β)
+        ( h)
+        ( cone-cartesian-hom-arrow g h β)
+        ( is-cartesian-cartesian-hom-arrow g h β)
+        ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow))
+
+  abstract
+    inv-coherence-triangle-hom-arrow-lift-map-codomain-cartesian-hom-arrow :
+      coherence-triangle-hom-arrow' f g h
+        ( α)
+        ( hom-arrow-cartesian-hom-arrow g h β)
+        ( hom-arrow-lift-map-codomain-cartesian-hom-arrow)
+    inv-coherence-triangle-hom-arrow-lift-map-codomain-cartesian-hom-arrow =
+      ( htpy-horizontal-map-gap-is-pullback
+          ( map-codomain-cartesian-hom-arrow g h β)
+          ( h)
+          ( cone-cartesian-hom-arrow g h β)
+          ( is-cartesian-cartesian-hom-arrow g h β)
+          ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow)) ,
+      ( H) ,
+      ( ( ap-concat-htpy'
+          ( ( h) ·l
+            ( htpy-horizontal-map-gap-is-pullback
+              ( map-codomain-cartesian-hom-arrow g h β)
+              ( h)
+              ( cone-cartesian-hom-arrow g h β)
+              ( pr2 β)
+              ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow)))
+          ( ap-concat-htpy'
+            ( coh-cartesian-hom-arrow g h β ·r
+              map-domain-hom-arrow-lift-map-codomain-cartesian-hom-arrow)
+            ( left-whisker-inv-htpy
+              ( map-codomain-cartesian-hom-arrow g h β)
+              ( htpy-vertical-map-gap-is-pullback
+                ( map-codomain-cartesian-hom-arrow g h β)
+                ( h)
+                ( cone-cartesian-hom-arrow g h β)
+                ( pr2 β)
+                ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow))))) ∙h
+        ( assoc-htpy
+          ( inv-htpy
+            ( ( map-codomain-cartesian-hom-arrow g h β) ·l
+              ( htpy-vertical-map-gap-is-pullback
+                ( map-codomain-cartesian-hom-arrow g h β)
+                ( h)
+                ( cone-cartesian-hom-arrow g h β)
+                ( pr2 β)
+                ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow))))
+          ( coh-cartesian-hom-arrow g h β ·r
+            map-domain-hom-arrow-lift-map-codomain-cartesian-hom-arrow)
+          ( ( h) ·l
+            ( htpy-horizontal-map-gap-is-pullback
+              ( map-codomain-cartesian-hom-arrow g h β)
+              ( h)
+              ( cone-cartesian-hom-arrow g h β)
+              ( pr2 β)
+              ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow)))) ∙h
+        ( inv-htpy-left-transpose-htpy-concat
+          ( ( map-codomain-cartesian-hom-arrow g h β) ·l
+            ( htpy-vertical-map-gap-is-pullback
+              ( map-codomain-cartesian-hom-arrow g h β)
+              ( h)
+              ( cone-cartesian-hom-arrow g h β)
+              ( pr2 β)
+              ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow)))
+          ( H ·r f ∙h coh-hom-arrow f h α)
+          ( ( coh-cartesian-hom-arrow g h β ·r
+              map-domain-hom-arrow-lift-map-codomain-cartesian-hom-arrow) ∙h
+            ( h) ·l
+            ( htpy-horizontal-map-gap-is-pullback
+              ( map-codomain-cartesian-hom-arrow g h β)
+              ( h)
+              ( cone-cartesian-hom-arrow g h β)
+              ( is-cartesian-cartesian-hom-arrow g h β)
+              ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow)))
+          ( inv-htpy
+            ( coh-htpy-cone-gap-is-pullback
+              ( map-codomain-cartesian-hom-arrow g h β)
+              ( h)
+              ( cone-cartesian-hom-arrow g h β)
+              ( is-cartesian-cartesian-hom-arrow g h β)
+              ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow)))))
+
+  coherence-triangle-hom-arrow-lift-map-codomain-cartesian-hom-arrow :
+    coherence-triangle-hom-arrow f g h
+      ( α)
+      ( hom-arrow-cartesian-hom-arrow g h β)
+      ( hom-arrow-lift-map-codomain-cartesian-hom-arrow)
+  coherence-triangle-hom-arrow-lift-map-codomain-cartesian-hom-arrow =
+    inv-htpy-hom-arrow f h
+      ( comp-hom-arrow f g h
+        ( hom-arrow-cartesian-hom-arrow g h β)
+        ( hom-arrow-lift-map-codomain-cartesian-hom-arrow))
+      ( α)
+      ( inv-coherence-triangle-hom-arrow-lift-map-codomain-cartesian-hom-arrow)
+```
+
+Now, if `α` was cartesian to begin with, the lift is also.
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (f : A → A') (g : B → B') (h : C → C')
+  (β : cartesian-hom-arrow g h)
+  (α : cartesian-hom-arrow f h)
+  (i : A' → B')
+  (H :
+    coherence-triangle-maps'
+      ( map-codomain-cartesian-hom-arrow f h α)
+      ( map-codomain-cartesian-hom-arrow g h β)
+      ( i))
+  where
+
+  abstract
+    is-cartesian-cartesian-hom-arrow-lift-map-codomain-cartesian-hom-arrow :
+      is-cartesian-hom-arrow f g
+        ( hom-arrow-lift-map-codomain-cartesian-hom-arrow f g h
+          ( β)
+          ( hom-arrow-cartesian-hom-arrow f h α)
+          ( i)
+          ( H))
+    is-cartesian-cartesian-hom-arrow-lift-map-codomain-cartesian-hom-arrow =
+      is-cartesian-top-cartesian-hom-arrow-triangle' f g h
+        ( hom-arrow-lift-map-codomain-cartesian-hom-arrow f g h
+          ( β)
+          ( hom-arrow-cartesian-hom-arrow f h α)
+          ( i)
+          ( H))
+        ( α)
+        ( β)
+        ( inv-coherence-triangle-hom-arrow-lift-map-codomain-cartesian-hom-arrow
+          ( f) g h β (hom-arrow-cartesian-hom-arrow f h α) i H)
+
+  cartesian-hom-arrow-lift-map-codomain-cartesian-hom-arrow :
+    cartesian-hom-arrow f g
+  cartesian-hom-arrow-lift-map-codomain-cartesian-hom-arrow =
+    ( hom-arrow-lift-map-codomain-cartesian-hom-arrow
+      ( f) g h β (hom-arrow-cartesian-hom-arrow f h α) i H) ,
+    ( is-cartesian-cartesian-hom-arrow-lift-map-codomain-cartesian-hom-arrow)
+```
diff --git a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
index 3d1e32d549..2302731798 100644
--- a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
@@ -29,9 +29,11 @@ open import foundation.morphisms-arrows
 open import foundation.precomposition-functions
 open import foundation.propositions
 open import foundation.pullbacks
+open import foundation.raising-universe-levels
 open import foundation.sections
 open import foundation.structure-identity-principle
 open import foundation.transport-along-identifications
+open import foundation.unit-type
 open import foundation.universe-levels
 open import foundation.whiskering-homotopies-composition
 
@@ -158,6 +160,45 @@ module _
   is-cartesian-cartesian-natural-transformation-polynomial-endofunctor = pr2 α
 ```
 
+### The criterion of being cartesian at terminal maps
+
+```agda
+module _
+  {l1 l2 l3 l4 l : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : natural-transformation-polynomial-endofunctor l 𝑃 𝑄)
+  where
+
+  is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor :
+    UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ lsuc l)
+  is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor =
+    {X : UU l} →
+    is-cartesian-hom-arrow
+      ( map-polynomial-endofunctor 𝑃 (raise-terminal-map X))
+      ( map-polynomial-endofunctor 𝑄 (raise-terminal-map X))
+      ( hom-arrow-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α
+        ( raise-terminal-map X))
+
+  is-prop-is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor :
+    is-prop
+      is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor
+  is-prop-is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor =
+    is-prop-implicit-Π
+      ( λ X →
+        is-prop-is-cartesian-hom-arrow
+          ( map-polynomial-endofunctor 𝑃 (raise-terminal-map X))
+          ( map-polynomial-endofunctor 𝑄 (raise-terminal-map X))
+          ( hom-arrow-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α
+            ( raise-terminal-map X)))
+
+  is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor-Prop :
+    Prop (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ lsuc l)
+  is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor-Prop =
+    ( is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor ,
+      is-prop-is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor)
+```
+
 ### The associated family of cartesian morphisms of arrows
 
 ```agda
@@ -236,3 +277,82 @@ module _
     ( comp-natural-transformation-polynomial-endofunctor 𝑃 𝑄 𝑅 β' α' ,
       is-cartesian-comp-cartesian-natural-transformation-polynomial-endofunctor)
 ```
+
+### A natural transformation is cartesian if and only if it is cartesian at terminal maps
+
+**Proof.** One direction is trivial. For the other direction, given a natural
+transformation of polynomial endofunctors $α : 𝑃 ⇒ 𝑄$ and an arbitrary function
+$f : X → Y$, we have a morphism of arrows in the slice above $α_{*}$:
+
+```text
+         αX
+  𝑃X -----------> 𝑄X   →
+   \ ⌟ →   𝑃Y ---- \ ----> 𝑄Y
+    \     / ⌟   αY  \     /
+     \   /           \   /
+      ∨ ∨             ∨ ∨
+       𝑃* -----------> 𝑄*
+              α*
+```
+
+and so by the right-cancellation property of cartesian squares the naturality
+square at $f$ is cartesian. ∎
+
+```agda
+module _
+  {l1 l2 l3 l4 l : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : natural-transformation-polynomial-endofunctor l 𝑃 𝑄)
+  (let α₀ = type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α)
+  where
+
+  is-cartesian-at-terminal-map-is-cartesian-natural-transformation-polynomial-endofunctor :
+    is-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α →
+    is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor
+      𝑃 𝑄 α
+  is-cartesian-at-terminal-map-is-cartesian-natural-transformation-polynomial-endofunctor
+    H {X} =
+    H (raise-terminal-map X)
+
+  abstract
+    is-cartesian-is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor :
+      is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor
+        𝑃 𝑄 α →
+      is-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α
+    is-cartesian-is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor
+      H {X} {Y} f =
+      is-pullback-top-square-vertical-triangle
+        α₀
+        ( map-polynomial-endofunctor 𝑄 (raise-terminal-map Y))
+        ( map-polynomial-endofunctor 𝑄 f)
+        ( map-polynomial-endofunctor 𝑄 (raise-terminal-map X))
+        ( cone-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α
+          ( raise-terminal-map Y))
+        ( cone-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α
+          ( f))
+        ( cone-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α
+          ( raise-terminal-map X))
+        ( refl-htpy)
+        ( refl-htpy ,
+          refl-htpy ,
+          ( λ (a , h) →
+            equational-reasoning
+            pr2 α (raise-terminal-map Y) (a , f ∘ h) ∙ ap (tot (λ x₁ f₁ a → map-raise star)) (pr2 α f (a , h)) ∙ refl
+            ＝
+              pr2 α (raise-terminal-map Y) (a , f ∘ h) ∙
+              ap (λ z → pair (pr1 z) (λ z₁ → map-raise star)) (pr2 α f (a , h))
+              by right-unit
+            ＝ {!   !} by {!   !}
+            ＝ pr2 α (raise-terminal-map X) (a , h) by {!   !})) -- these are two equalities in the type (Σ 𝑄₀ , 1), so they only need to be equal on first components.
+        ( H {Y})
+        ( H {X})
+```
+
+<!-- TODO: use uniqueness of cartesian lifts -->
+
+This is mentioned as Remark 2.1.4 in {{#cite GHK22}}.
+
+## References
+
+{{#bibliography}}

From 702071c05145e6f035c4e909c270b5ded003b5b3 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Tue, 16 Sep 2025 01:37:32 +0200
Subject: [PATCH 25/41] some parentheses

---
 src/foundation/homotopies.lagda.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/foundation/homotopies.lagda.md b/src/foundation/homotopies.lagda.md
index 88a2ba84ea..c61913e927 100644
--- a/src/foundation/homotopies.lagda.md
+++ b/src/foundation/homotopies.lagda.md
@@ -225,7 +225,7 @@ module _
     is-equiv-map-Π-is-fiberwise-equiv
       ( λ x → is-equiv-left-transpose-eq-concat (H x) (K x) (L x))
 
-  equiv-left-transpose-htpy-concat : (H ∙h K ~ L) ≃ (K ~ (inv-htpy H ∙h L))
+  equiv-left-transpose-htpy-concat : (H ∙h K ~ L) ≃ (K ~ inv-htpy H ∙h L)
   pr1 equiv-left-transpose-htpy-concat = left-transpose-htpy-concat H K L
   pr2 equiv-left-transpose-htpy-concat = is-equiv-left-transpose-htpy-concat
 
@@ -235,7 +235,7 @@ module _
     is-equiv-map-Π-is-fiberwise-equiv
       ( λ x → is-equiv-right-transpose-eq-concat (H x) (K x) (L x))
 
-  equiv-right-transpose-htpy-concat : ((H ∙h K) ~ L) ≃ (H ~ (L ∙h (inv-htpy K)))
+  equiv-right-transpose-htpy-concat : (H ∙h K ~ L) ≃ (H ~ L ∙h inv-htpy K)
   pr1 equiv-right-transpose-htpy-concat = right-transpose-htpy-concat H K L
   pr2 equiv-right-transpose-htpy-concat = is-equiv-right-transpose-htpy-concat
 

From f31bed5b9607e5db4f66e4c7d4b2887c9bddd1f7 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Tue, 16 Sep 2025 11:59:12 +0200
Subject: [PATCH 26/41] save some space

---
 ...operty-cartesian-morphisms-arrows.lagda.md | 197 ++++++++----------
 1 file changed, 82 insertions(+), 115 deletions(-)

diff --git a/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md b/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
index d379555370..fdf97ae8bf 100644
--- a/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
+++ b/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
@@ -148,11 +148,11 @@ module _
   (α : hom-arrow f h) (β : hom-arrow g h)
   where
 
-  family-fiber-lift-codomain-lift-hom-arrow :
+  is-lift-hom-arrow-of-lift-codomain-hom-arrow :
     lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α) →
     (A → B) → UU (l1 ⊔ l4 ⊔ l5 ⊔ l6)
-  family-fiber-lift-codomain-lift-hom-arrow (i , I) j =
-    Σ (coherence-hom-arrow f g j i)
+  is-lift-hom-arrow-of-lift-codomain-hom-arrow (i , I) j =
+    Σ ( coherence-hom-arrow f g j i)
       ( λ H →
         Σ ( coherence-triangle-maps
             ( map-domain-hom-arrow f h α)
@@ -163,21 +163,14 @@ module _
               ( comp-hom-arrow f g h β (j , i , H))
               ( J)
               ( I)))
-
-  fiber-lift-codomain-lift-hom-arrow :
-    lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α) →
-    UU (l1 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6)
-  fiber-lift-codomain-lift-hom-arrow i =
-    Σ (A → B) (family-fiber-lift-codomain-lift-hom-arrow i)
 ```
 
 > The explicit equivalence remains to be written out.
 
 ### Uniqueness of lifts of cartesian morphisms
 
-Given a cartesian morphism of arrows, then lifts are unique.
-
-<!-- The following equivalence uses function extensionality in many places it does not need to -->
+Given a cartesian morphism of arrows, then lifts along lifts in the codomain are
+unique.
 
 ```agda
 module _
@@ -185,122 +178,96 @@ module _
   {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
   (f : A → A') (g : B → B') (h : C → C')
   (α : hom-arrow f h) (β : hom-arrow g h)
+  (let β₁ = map-codomain-hom-arrow g h β)
+  (let Hβ = coh-hom-arrow g h β)
+  (let Hα = coh-hom-arrow f h α)
   ((i , I) : lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α))
   where
 
-  abstract
-    equiv-htpy-cone-family-fiber-lift-codomain-lift-hom-arrow :
-      (j : A → B) →
-      family-fiber-lift-codomain-lift-hom-arrow f g h α β (i , I) j ≃
-      htpy-cone (map-codomain-hom-arrow g h β) h
-        ( g ∘ j ,
-          map-domain-hom-arrow g h β ∘ j ,
-          coh-hom-arrow g h β ·r j)
-        ( i ∘ f ,
-          map-domain-hom-arrow f h α ,
-          inv-htpy I ·r f ∙h coh-hom-arrow f h α)
-    equiv-htpy-cone-family-fiber-lift-codomain-lift-hom-arrow j =
-      equiv-Σ _
-        ( equiv-inv-htpy (i ∘ f) (g ∘ j))
-        ( λ H →
-          equiv-Σ _
-            ( equiv-inv-htpy (pr1 α) (pr1 β ∘ j))
-            ( λ J →
-              equivalence-reasoning
-              ( ( coh-hom-arrow f h α ∙h h ·l J) ~
-                ( I ·r f) ∙h
-                ( ( map-codomain-hom-arrow g h β ·l H) ∙h
-                  ( coh-hom-arrow g h β ·r j)))
-              ≃ ( ( I ·r f) ∙h
-                  ( ( map-codomain-hom-arrow g h β ·l H) ∙h
-                    ( coh-hom-arrow g h β ·r j)) ~
-                  ( coh-hom-arrow f h α ∙h h ·l J))
-                by equiv-inv-htpy _ _
-              ≃ ( ( map-codomain-hom-arrow g h β ·l H) ∙h
-                  ( coh-hom-arrow g h β ·r j) ~
-                  ( inv-htpy I ·r f) ∙h (coh-hom-arrow f h α ∙h h ·l J))
-                by
-                  equiv-left-transpose-htpy-concat
-                    ( I ·r f)
-                    ( ( map-codomain-hom-arrow g h β ·l H) ∙h
-                      ( coh-hom-arrow g h β ·r j))
-                    ( coh-hom-arrow f h α ∙h h ·l J)
-              ≃ ( ( coh-hom-arrow g h β ·r j) ~
-                  ( inv-htpy (map-codomain-hom-arrow g h β ·l H)) ∙h
-                  ( inv-htpy I ·r f ∙h (coh-hom-arrow f h α ∙h h ·l J)))
-                by
-                  equiv-left-transpose-htpy-concat
-                    ( map-codomain-hom-arrow g h β ·l H)
-                    ( coh-hom-arrow g h β ·r j)
-                    ( inv-htpy I ·r f ∙h (coh-hom-arrow f h α ∙h h ·l J))
-              ≃ ( ( coh-hom-arrow g h β ·r j) ~
-                  ( map-codomain-hom-arrow g h β ·l (inv-htpy H)) ∙h
-                  ( inv-htpy I ·r f ∙h (coh-hom-arrow f h α ∙h h ·l J)))
-                by
-                  equiv-tr
-                    ( λ Q →
-                      coh-hom-arrow g h β ·r j ~
-                      Q ∙h (inv-htpy I ·r f ∙h (coh-hom-arrow f h α ∙h h ·l J)))
-                    ( inv
-                      ( eq-htpy
-                        ( left-whisker-inv-htpy
-                          ( map-codomain-hom-arrow g h β) H)))
-              ≃ ( ( coh-hom-arrow g h β ·r j) ~
-                  ( ( map-codomain-hom-arrow g h β ·l (inv-htpy H)) ∙h
-                    ( inv-htpy I ·r f ∙h (coh-hom-arrow f h α))) ∙h
-                  ( h ·l J))
-                by
-                  equiv-tr
-                    ( coh-hom-arrow g h β ·r j ~_)
-                    ( ( ap
-                        ( map-codomain-hom-arrow g h β ·l (inv-htpy H) ∙h_)
-                        ( eq-htpy
-                          ( inv-htpy-assoc-htpy
-                            ( inv-htpy I ·r f)
-                            ( coh-hom-arrow f h α)
-                            ( h ·l J)))) ∙
+  equiv-htpy-cone-is-lift-hom-arrow-of-lift-codomain-hom-arrow :
+    (j : A → B) →
+    is-lift-hom-arrow-of-lift-codomain-hom-arrow f g h α β (i , I) j ≃
+    htpy-cone β₁ h
+      ( g ∘ j , map-domain-hom-arrow g h β ∘ j , Hβ ·r j)
+      ( i ∘ f , map-domain-hom-arrow f h α , inv-htpy I ·r f ∙h Hα)
+  equiv-htpy-cone-is-lift-hom-arrow-of-lift-codomain-hom-arrow j =
+    equiv-Σ _
+      ( equiv-inv-htpy (i ∘ f) (g ∘ j))
+      ( λ H →
+        equiv-Σ _
+          ( equiv-inv-htpy
+            ( map-domain-hom-arrow f h α)
+            ( map-domain-hom-arrow g h β ∘ j))
+          ( λ J →
+            equivalence-reasoning
+            ( Hα ∙h h ·l J ~ I ·r f ∙h (β₁ ·l H ∙h Hβ ·r j))
+            ≃ ( I ·r f ∙h (β₁ ·l H ∙h Hβ ·r j) ~ Hα ∙h h ·l J)
+              by equiv-inv-htpy (Hα ∙h h ·l J) (I ·r f ∙h (β₁ ·l H ∙h Hβ ·r j))
+            ≃ ( β₁ ·l H ∙h Hβ ·r j ~ inv-htpy I ·r f ∙h (Hα ∙h h ·l J))
+              by
+                equiv-left-transpose-htpy-concat
+                  ( I ·r f)
+                  ( β₁ ·l H ∙h Hβ ·r j)
+                  ( Hα ∙h h ·l J)
+            ≃ ( Hβ ·r j ~
+                inv-htpy (β₁ ·l H) ∙h (inv-htpy I ·r f ∙h (Hα ∙h h ·l J)))
+              by
+                equiv-left-transpose-htpy-concat
+                  ( β₁ ·l H)
+                  ( Hβ ·r j)
+                  ( inv-htpy I ·r f ∙h (Hα ∙h h ·l J))
+            ≃ ( Hβ ·r j ~
+                β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h (Hα ∙h h ·l J)))
+              by
+                equiv-tr
+                  ( λ Q → Hβ ·r j ~ Q ∙h (inv-htpy I ·r f ∙h (Hα ∙h h ·l J)))
+                  ( inv (eq-htpy (left-whisker-inv-htpy β₁ H)))
+            ≃ ( Hβ ·r j ~
+                ( β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα)) ∙h h ·l J)
+              by
+                equiv-tr
+                  ( Hβ ·r j ~_)
+                  ( ( ap
+                      ( β₁ ·l (inv-htpy H) ∙h_)
                       ( eq-htpy
-                        ( inv-htpy-assoc-htpy
-                          ( map-codomain-hom-arrow g h β ·l (inv-htpy H))
-                          ( inv-htpy I ·r f ∙h (coh-hom-arrow f h α))
-                          ( h ·l J))))
-              ≃ ( ( coh-hom-arrow g h β ·r j) ∙h (inv-htpy (h ·l J)) ~
-                  ( map-codomain-hom-arrow g h β ·l (inv-htpy H)) ∙h
-                  ( (inv-htpy I ·r f) ∙h (coh-hom-arrow f h α)))
-                by
-                  equiv-right-transpose-htpy-concat'
-                    ( coh-hom-arrow g h β ·r j)
-                    ( ( map-codomain-hom-arrow g h β ·l (inv-htpy H)) ∙h
-                      ( inv-htpy I ·r f ∙h (coh-hom-arrow f h α)))
-                    ( h ·l J)
-              ≃ ( ( coh-hom-arrow g h β ·r j) ∙h (h ·l inv-htpy J) ~
-                  ( map-codomain-hom-arrow g h β ·l inv-htpy H) ∙h
-                  ( ( inv-htpy I ·r f) ∙h (coh-hom-arrow f h α)))
-                by
-                  equiv-tr
-                    ( λ Q →
-                      ( coh-hom-arrow g h β ·r j) ∙h Q ~
-                      ( map-codomain-hom-arrow g h β ·l inv-htpy H) ∙h
-                      ( ( inv-htpy I ·r f) ∙h (coh-hom-arrow f h α)))
-                    ( inv (eq-htpy (left-whisker-inv-htpy h J)))))
+                        ( inv-htpy-assoc-htpy (inv-htpy I ·r f) Hα (h ·l J)))) ∙
+                    ( eq-htpy
+                      ( inv-htpy-assoc-htpy
+                        ( β₁ ·l (inv-htpy H))
+                        ( inv-htpy I ·r f ∙h Hα)
+                        ( h ·l J))))
+            ≃ ( Hβ ·r j ∙h inv-htpy (h ·l J) ~
+                β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))
+              by
+                equiv-right-transpose-htpy-concat'
+                  ( Hβ ·r j)
+                  ( β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))
+                  ( h ·l J)
+            ≃ ( Hβ ·r j ∙h h ·l inv-htpy J ~
+                β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))
+              by
+                equiv-tr
+                  ( λ Q →
+                    Hβ ·r j ∙h Q ~ β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))
+                  ( inv (eq-htpy (left-whisker-inv-htpy h J)))))
 
   abstract
     uniqueness-lift-map-codomain-is-cartesian-hom-arrow :
       is-cartesian-hom-arrow g h β →
-      is-torsorial (family-fiber-lift-codomain-lift-hom-arrow f g h α β (i , I))
+      is-torsorial
+        ( is-lift-hom-arrow-of-lift-codomain-hom-arrow f g h α β (i , I))
     uniqueness-lift-map-codomain-is-cartesian-hom-arrow H =
       is-contr-equiv _
-        ( equiv-tot equiv-htpy-cone-family-fiber-lift-codomain-lift-hom-arrow)
-        ( uniqueness-universal-property-pullback
-          ( map-codomain-hom-arrow g h β)
-          ( h)
+        ( equiv-tot
+          ( equiv-htpy-cone-is-lift-hom-arrow-of-lift-codomain-hom-arrow))
+        ( uniqueness-universal-property-pullback β₁ h
           ( cone-hom-arrow g h β)
           ( up-pullback-cartesian-hom-arrow g h (β , H))
           ( A)
-          ( _))
+          ( i ∘ f , map-domain-hom-arrow f h α , inv-htpy I ·r f ∙h Hα))
 ```
 
-<!-- Theis uniqueness proof also constructs an explicit lift, making the entire next section entirely useless -->
+<!-- This uniqueness proof also constructs an explicit lift, making the entire next section purposeless -->
 
 > What follows is another computation of the same. Should be deleted.
 
@@ -338,7 +305,7 @@ module _
                     ( K)
                     ( H))))
     uniqueness-lift-map-codomain-cartesian-hom-arrow' =
-      is-contr-equiv ((Σ (A → B) λ j → htpy-cone (map-codomain-cartesian-hom-arrow g h β) h ((λ z → g (j z)) , (λ z → pr1 (pr1 β) (j z)) , (λ x → pr2 (pr2 (pr1 β)) (j x))) ((λ z → i (f z)) , pr1 α , H ·r f ∙h coh-hom-arrow f h α)))
+      is-contr-equiv _
         ( equiv-tot
           ( λ j →
             equiv-Σ _
@@ -394,7 +361,7 @@ module _
           ( cone-cartesian-hom-arrow g h β)
           ( up-pullback-cartesian-hom-arrow g h β)
           ( A)
-          ( cone-hom-arrow-lift-map-codomain-cartesian-hom-arrow f g h β α i H))
+          ( i ∘ f , pr1 α , H ·r f ∙h coh-hom-arrow f h α))
 ```
 
 ### Lifting cartesian morphisms along lifts of the codomain

From 8138263004a7fdcd599a80ad713c0f741110ab62 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Tue, 16 Sep 2025 13:56:49 +0200
Subject: [PATCH 27/41] `compute-fiber-lift-codomain-lift-hom-arrow`

---
 ...e-arithmetic-dependent-pair-types.lagda.md |  2 +-
 ...operty-cartesian-morphisms-arrows.lagda.md | 84 ++++++++++++++++++-
 2 files changed, 83 insertions(+), 3 deletions(-)

diff --git a/src/foundation/type-arithmetic-dependent-pair-types.lagda.md b/src/foundation/type-arithmetic-dependent-pair-types.lagda.md
index 022b2dc3a8..b4440ce31d 100644
--- a/src/foundation/type-arithmetic-dependent-pair-types.lagda.md
+++ b/src/foundation/type-arithmetic-dependent-pair-types.lagda.md
@@ -308,7 +308,7 @@ module _
 
   interchange-Σ-Σ-Σ :
     Σ A (λ x → Σ (B x) (λ y → Σ (C x) (D x y))) ≃
-    Σ A (λ x → Σ (C x) (λ z → Σ (B x) λ y → D x y z))
+    Σ A (λ x → Σ (C x) (λ z → Σ (B x) (λ y → D x y z)))
   interchange-Σ-Σ-Σ =
     associative-Σ' A C (λ x z → Σ (B x) λ y → D x y z) ∘e
     interchange-Σ-Σ ∘e
diff --git a/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md b/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
index fdf97ae8bf..29f7575d70 100644
--- a/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
+++ b/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
@@ -15,10 +15,12 @@ open import foundation.cones-over-cospan-diagrams
 open import foundation.contractible-types
 open import foundation.coproducts-pullbacks
 open import foundation.dependent-pair-types
+open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.dependent-products-pullbacks
 open import foundation.dependent-sums-pullbacks
 open import foundation.diagonal-maps-cartesian-products-of-types
 open import foundation.equivalences
+open import foundation.homotopy-induction
 open import foundation.fibers-of-maps
 open import foundation.function-extensionality
 open import foundation.function-types
@@ -146,6 +148,7 @@ module _
   {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
   (f : A → A') (g : B → B') (h : C → C')
   (α : hom-arrow f h) (β : hom-arrow g h)
+  ( let β₁ = map-codomain-hom-arrow g h β)
   where
 
   is-lift-hom-arrow-of-lift-codomain-hom-arrow :
@@ -163,9 +166,86 @@ module _
               ( comp-hom-arrow f g h β (j , i , H))
               ( J)
               ( I)))
-```
 
-> The explicit equivalence remains to be written out.
+  lift-hom-arrow-of-lift-codomain-hom-arrow :
+    lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α) →
+    UU (l1 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6)
+  lift-hom-arrow-of-lift-codomain-hom-arrow i =
+    Σ (A → B) (is-lift-hom-arrow-of-lift-codomain-hom-arrow i)
+
+  compute-fiber-lift-codomain-lift-hom-arrow :
+    (δ : lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α)) →
+    fiber (lift-codomain-lift-hom-arrow f g h α β) δ ≃
+    lift-hom-arrow-of-lift-codomain-hom-arrow δ
+  compute-fiber-lift-codomain-lift-hom-arrow (δ , Hδ) =
+    equivalence-reasoning
+    Σ ( lift-hom-arrow f g h α β)
+      ( λ γ → lift-codomain-lift-hom-arrow f g h α β γ ＝ (δ , Hδ))
+    ≃ Σ ( lift-hom-arrow f g h α β)
+        ( λ γ →
+          Σ ( map-codomain-lift-hom-arrow f g h α β γ ~ δ)
+            ( λ Hi → coh-codomain-lift-hom-arrow f g h α β γ ∙h β₁ ·l Hi ~ Hδ))
+      by
+        equiv-tot
+          ( λ γ →
+            extensionality-lift
+              ( map-codomain-hom-arrow g h β)
+              ( map-codomain-hom-arrow f h α)
+              ( lift-codomain-lift-hom-arrow f g h α β γ)
+              ( δ , Hδ))
+    ≃ Σ ( Σ (A' → B') (_~ δ))
+        ( λ (i , Hi) →
+          Σ ( Σ ( coherence-triangle-maps
+                  ( map-codomain-hom-arrow f h α)
+                  ( map-codomain-hom-arrow g h β)
+                  ( i))
+                ( λ I → I ∙h β₁ ·l Hi ~ Hδ))
+            ( λ (I , HI) → lift-hom-arrow-of-lift-codomain-hom-arrow (i , I)))
+      by reassociate
+    ≃ Σ ( Σ ( coherence-triangle-maps
+              ( map-codomain-hom-arrow f h α)
+              ( map-codomain-hom-arrow g h β)
+              ( δ))
+            ( λ I → I ∙h β₁ ·l refl-htpy ~ Hδ))
+        ( λ (I , HI) → lift-hom-arrow-of-lift-codomain-hom-arrow (δ , I))
+      by left-unit-law-Σ-is-contr (is-torsorial-htpy' δ) (δ , refl-htpy)
+    ≃ Σ ( Σ ( coherence-triangle-maps
+              ( map-codomain-hom-arrow f h α)
+              ( map-codomain-hom-arrow g h β)
+              ( δ))
+            ( _~ Hδ))
+        ( λ (I , HI) → lift-hom-arrow-of-lift-codomain-hom-arrow (δ , I))
+      by
+        equiv-Σ-equiv-base
+          ( λ (I , HI) → lift-hom-arrow-of-lift-codomain-hom-arrow (δ , I))
+          ( equiv-tot (λ I → equiv-concat-htpy inv-htpy-right-unit-htpy Hδ))
+    ≃ lift-hom-arrow-of-lift-codomain-hom-arrow (δ , Hδ)
+      by left-unit-law-Σ-is-contr (is-torsorial-htpy' Hδ) (Hδ , refl-htpy)
+    where
+      reassociate :
+        Σ ( lift-hom-arrow f g h α β)
+        ( λ γ →
+          Σ ( map-codomain-lift-hom-arrow f g h α β γ ~ δ)
+            ( λ Hi →
+              coh-codomain-lift-hom-arrow f g h α β γ ∙h β₁ ·l Hi ~ Hδ)) ≃
+        Σ ( Σ (A' → B') (_~ δ))
+          ( λ (i , Hi) →
+            Σ ( Σ ( coherence-triangle-maps
+                    ( map-codomain-hom-arrow f h α)
+                    ( map-codomain-hom-arrow g h β)
+                    ( i))
+                  ( λ I → I ∙h β₁ ·l Hi ~ Hδ))
+                ( λ (I , HI) →
+                  lift-hom-arrow-of-lift-codomain-hom-arrow (i , I)))
+      reassociate =
+        ( ( λ (((γ₀ , γ₁ , Hγ) , (Γ₀ , Γ₁ , HΓ)) , (Hi , HI)) →
+            ( (γ₁ , Hi) , (Γ₁ , HI) , γ₀ , Hγ , Γ₀ , HΓ)) ,
+          ( is-equiv-is-invertible
+            ( λ ((γ₁ , Hi) , (Γ₁ , HI) , γ₀ , Hγ , Γ₀ , HΓ) →
+              ((γ₀ , γ₁ , Hγ) , (Γ₀ , Γ₁ , HΓ)) , (Hi , HI))
+            ( refl-htpy)
+            ( refl-htpy)))
+```
 
 ### Uniqueness of lifts of cartesian morphisms
 

From f8a2df68b84981b498188e63914ec48b637f9e9a Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Tue, 16 Sep 2025 15:36:25 +0200
Subject: [PATCH 28/41] equivalence between up-cartesian, having unique lifts,
 and cartesianness

---
 .../universal-property-pullbacks.lagda.md     |  12 +
 .../lifts-morphisms-arrows.lagda.md           | 227 +++++++++
 ...operty-cartesian-morphisms-arrows.lagda.md | 460 +++++-------------
 3 files changed, 371 insertions(+), 328 deletions(-)

diff --git a/src/foundation-core/universal-property-pullbacks.lagda.md b/src/foundation-core/universal-property-pullbacks.lagda.md
index 8ec97f1f27..298af3f46c 100644
--- a/src/foundation-core/universal-property-pullbacks.lagda.md
+++ b/src/foundation-core/universal-property-pullbacks.lagda.md
@@ -219,6 +219,18 @@ module _
         ( Σ (C' → C) (λ h → cone-map f g c h ＝ c'))
         ( equiv-tot (λ h → extensionality-cone f g (cone-map f g c h) c'))
         ( is-contr-map-is-equiv (up C') c')
+
+  abstract
+    universal-property-pullback-uniqueness :
+      ( {l5 : Level} (C' : UU l5) (c' : cone f g C') →
+        is-contr (Σ (C' → C) (λ h → htpy-cone f g (cone-map f g c h) c'))) →
+      universal-property-pullback f g c
+    universal-property-pullback-uniqueness H C' =
+      is-equiv-is-contr-map
+        ( λ c' →
+          is-contr-equiv _
+            ( equiv-tot (λ h → extensionality-cone f g (cone-map f g c h) c'))
+            ( H C' c'))
 ```
 
 ## Table of files about pullbacks
diff --git a/src/foundation/lifts-morphisms-arrows.lagda.md b/src/foundation/lifts-morphisms-arrows.lagda.md
index aa2f4f318a..ac8487736a 100644
--- a/src/foundation/lifts-morphisms-arrows.lagda.md
+++ b/src/foundation/lifts-morphisms-arrows.lagda.md
@@ -7,12 +7,27 @@ module foundation.lifts-morphisms-arrows where
 <details><summary>Imports</summary>
 
 ```agda
+open import foundation.action-on-identifications-functions
 open import foundation.commuting-triangles-of-maps
 open import foundation.commuting-triangles-of-morphisms-arrows
+open import foundation.cones-over-cospan-diagrams
 open import foundation.dependent-pair-types
+open import foundation.equivalences
+open import foundation.fibers-of-maps
+open import foundation.function-extensionality
+open import foundation.function-types
+open import foundation.functoriality-dependent-pair-types
+open import foundation.homotopies
 open import foundation.homotopies-morphisms-arrows
+open import foundation.homotopy-induction
+open import foundation.identity-types
 open import foundation.morphisms-arrows
+open import foundation.transport-along-identifications
+open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
+open import foundation.whiskering-homotopies-composition
+
+open import orthogonal-factorization-systems.lifts-maps
 ```
 
 </details>
@@ -90,4 +105,216 @@ module _
       ( coh-codomain-lift-hom-arrow)
   coh-lift-hom-arrow =
     pr2 (pr2 coherence-triangle-lift-hom-arrow)
+
+  lift-codomain-lift-hom-arrow :
+    lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α)
+  lift-codomain-lift-hom-arrow =
+    ( map-codomain-lift-hom-arrow , coh-codomain-lift-hom-arrow)
+```
+
+### The type of lifts extending a lift of the codomain
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (f : A → A') (g : B → B') (h : C → C')
+  (α : hom-arrow f h) (β : hom-arrow g h)
+  ( let β₁ = map-codomain-hom-arrow g h β)
+  where
+
+  is-lift-hom-arrow-of-lift-codomain-hom-arrow :
+    lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α) →
+    (A → B) → UU (l1 ⊔ l4 ⊔ l5 ⊔ l6)
+  is-lift-hom-arrow-of-lift-codomain-hom-arrow (i , I) j =
+    Σ ( coherence-hom-arrow f g j i)
+      ( λ H →
+        Σ ( coherence-triangle-maps
+            ( map-domain-hom-arrow f h α)
+            ( map-domain-hom-arrow g h β)
+            ( j))
+          ( λ J →
+            coherence-htpy-hom-arrow f h α
+              ( comp-hom-arrow f g h β (j , i , H))
+              ( J)
+              ( I)))
+
+  lift-hom-arrow-of-lift-codomain-hom-arrow :
+    lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α) →
+    UU (l1 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6)
+  lift-hom-arrow-of-lift-codomain-hom-arrow i =
+    Σ (A → B) (is-lift-hom-arrow-of-lift-codomain-hom-arrow i)
+```
+
+## Properties
+
+### Computing the fibers of `lift-codomain-lift-hom-arrow`
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (f : A → A') (g : B → B') (h : C → C')
+  (α : hom-arrow f h) (β : hom-arrow g h)
+  ( let β₁ = map-codomain-hom-arrow g h β)
+  where
+
+  compute-fiber-lift-codomain-lift-hom-arrow :
+    (δ : lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α)) →
+    fiber (lift-codomain-lift-hom-arrow f g h α β) δ ≃
+    lift-hom-arrow-of-lift-codomain-hom-arrow f g h α β δ
+  compute-fiber-lift-codomain-lift-hom-arrow (δ , Hδ) =
+    equivalence-reasoning
+    Σ ( lift-hom-arrow f g h α β)
+      ( λ γ → lift-codomain-lift-hom-arrow f g h α β γ ＝ (δ , Hδ))
+    ≃ Σ ( lift-hom-arrow f g h α β)
+        ( λ γ →
+          Σ ( map-codomain-lift-hom-arrow f g h α β γ ~ δ)
+            ( λ Hi → coh-codomain-lift-hom-arrow f g h α β γ ∙h β₁ ·l Hi ~ Hδ))
+      by
+        equiv-tot
+          ( λ γ →
+            extensionality-lift
+              ( map-codomain-hom-arrow g h β)
+              ( map-codomain-hom-arrow f h α)
+              ( lift-codomain-lift-hom-arrow f g h α β γ)
+              ( δ , Hδ))
+    ≃ Σ ( Σ (A' → B') (_~ δ))
+        ( λ (i , Hi) →
+          Σ ( Σ ( coherence-triangle-maps
+                  ( map-codomain-hom-arrow f h α)
+                  ( map-codomain-hom-arrow g h β)
+                  ( i))
+                ( λ I → I ∙h β₁ ·l Hi ~ Hδ))
+            ( λ (I , HI) →
+              lift-hom-arrow-of-lift-codomain-hom-arrow f g h α β (i , I)))
+      by reassociate
+    ≃ Σ ( Σ ( coherence-triangle-maps
+              ( map-codomain-hom-arrow f h α)
+              ( map-codomain-hom-arrow g h β)
+              ( δ))
+            ( λ I → I ∙h β₁ ·l refl-htpy ~ Hδ))
+        ( λ (I , HI) →
+          lift-hom-arrow-of-lift-codomain-hom-arrow f g h α β (δ , I))
+      by left-unit-law-Σ-is-contr (is-torsorial-htpy' δ) (δ , refl-htpy)
+    ≃ Σ ( Σ ( coherence-triangle-maps
+              ( map-codomain-hom-arrow f h α)
+              ( map-codomain-hom-arrow g h β)
+              ( δ))
+            ( _~ Hδ))
+        ( λ (I , HI) →
+          lift-hom-arrow-of-lift-codomain-hom-arrow f g h α β (δ , I))
+      by
+        equiv-Σ-equiv-base
+          ( λ (I , HI) →
+            lift-hom-arrow-of-lift-codomain-hom-arrow f g h α β (δ , I))
+          ( equiv-tot (λ I → equiv-concat-htpy inv-htpy-right-unit-htpy Hδ))
+    ≃ lift-hom-arrow-of-lift-codomain-hom-arrow f g h α β (δ , Hδ)
+      by left-unit-law-Σ-is-contr (is-torsorial-htpy' Hδ) (Hδ , refl-htpy)
+    where
+      reassociate :
+        Σ ( lift-hom-arrow f g h α β)
+        ( λ γ →
+          Σ ( map-codomain-lift-hom-arrow f g h α β γ ~ δ)
+            ( λ Hi →
+              coh-codomain-lift-hom-arrow f g h α β γ ∙h β₁ ·l Hi ~ Hδ)) ≃
+        Σ ( Σ (A' → B') (_~ δ))
+          ( λ (i , Hi) →
+            Σ ( Σ ( coherence-triangle-maps
+                    ( map-codomain-hom-arrow f h α)
+                    ( map-codomain-hom-arrow g h β)
+                    ( i))
+                  ( λ I → I ∙h β₁ ·l Hi ~ Hδ))
+                ( λ (I , HI) →
+                  lift-hom-arrow-of-lift-codomain-hom-arrow f g h α β (i , I)))
+      reassociate =
+        ( ( λ (((γ₀ , γ₁ , Hγ) , (Γ₀ , Γ₁ , HΓ)) , (Hi , HI)) →
+            ( (γ₁ , Hi) , (Γ₁ , HI) , γ₀ , Hγ , Γ₀ , HΓ)) ,
+          ( is-equiv-is-invertible
+            ( λ ((γ₁ , Hi) , (Γ₁ , HI) , γ₀ , Hγ , Γ₀ , HΓ) →
+              ((γ₀ , γ₁ , Hγ) , (Γ₀ , Γ₁ , HΓ)) , (Hi , HI))
+            ( refl-htpy)
+            ( refl-htpy)))
+```
+
+### Computing the "is lift" predicate in terms of homotopies of cones
+
+```agda
+module _
+  {l1 l2 l3 l4 l5 l6 : Level}
+  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (f : A → A') (g : B → B') (h : C → C')
+  (α : hom-arrow f h) (β : hom-arrow g h)
+  (let β₁ = map-codomain-hom-arrow g h β)
+  (let Hβ = coh-hom-arrow g h β)
+  (let Hα = coh-hom-arrow f h α)
+  ((i , I) : lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α))
+  where
+
+  equiv-htpy-cone-is-lift-hom-arrow-of-lift-codomain-hom-arrow :
+    (j : A → B) →
+    is-lift-hom-arrow-of-lift-codomain-hom-arrow f g h α β (i , I) j ≃
+    htpy-cone β₁ h
+      ( g ∘ j , map-domain-hom-arrow g h β ∘ j , Hβ ·r j)
+      ( i ∘ f , map-domain-hom-arrow f h α , inv-htpy I ·r f ∙h Hα)
+  equiv-htpy-cone-is-lift-hom-arrow-of-lift-codomain-hom-arrow j =
+    equiv-Σ _
+      ( equiv-inv-htpy (i ∘ f) (g ∘ j))
+      ( λ H →
+        equiv-Σ _
+          ( equiv-inv-htpy
+            ( map-domain-hom-arrow f h α)
+            ( map-domain-hom-arrow g h β ∘ j))
+          ( λ J →
+            equivalence-reasoning
+            ( Hα ∙h h ·l J ~ I ·r f ∙h (β₁ ·l H ∙h Hβ ·r j))
+            ≃ ( I ·r f ∙h (β₁ ·l H ∙h Hβ ·r j) ~ Hα ∙h h ·l J)
+              by equiv-inv-htpy (Hα ∙h h ·l J) (I ·r f ∙h (β₁ ·l H ∙h Hβ ·r j))
+            ≃ ( β₁ ·l H ∙h Hβ ·r j ~ inv-htpy I ·r f ∙h (Hα ∙h h ·l J))
+              by
+                equiv-left-transpose-htpy-concat
+                  ( I ·r f)
+                  ( β₁ ·l H ∙h Hβ ·r j)
+                  ( Hα ∙h h ·l J)
+            ≃ ( Hβ ·r j ~
+                inv-htpy (β₁ ·l H) ∙h (inv-htpy I ·r f ∙h (Hα ∙h h ·l J)))
+              by
+                equiv-left-transpose-htpy-concat
+                  ( β₁ ·l H)
+                  ( Hβ ·r j)
+                  ( inv-htpy I ·r f ∙h (Hα ∙h h ·l J))
+            ≃ ( Hβ ·r j ~
+                β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h (Hα ∙h h ·l J)))
+              by
+                equiv-tr
+                  ( λ Q → Hβ ·r j ~ Q ∙h (inv-htpy I ·r f ∙h (Hα ∙h h ·l J)))
+                  ( inv (eq-htpy (left-whisker-inv-htpy β₁ H)))
+            ≃ ( Hβ ·r j ~
+                ( β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα)) ∙h h ·l J)
+              by
+                equiv-tr
+                  ( Hβ ·r j ~_)
+                  ( ( ap
+                      ( β₁ ·l (inv-htpy H) ∙h_)
+                      ( eq-htpy
+                        ( inv-htpy-assoc-htpy (inv-htpy I ·r f) Hα (h ·l J)))) ∙
+                    ( eq-htpy
+                      ( inv-htpy-assoc-htpy
+                        ( β₁ ·l (inv-htpy H))
+                        ( inv-htpy I ·r f ∙h Hα)
+                        ( h ·l J))))
+            ≃ ( Hβ ·r j ∙h inv-htpy (h ·l J) ~
+                β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))
+              by
+                equiv-right-transpose-htpy-concat'
+                  ( Hβ ·r j)
+                  ( β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))
+                  ( h ·l J)
+            ≃ ( Hβ ·r j ∙h h ·l inv-htpy J ~
+                β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))
+              by
+                equiv-tr
+                  ( λ Q →
+                    Hβ ·r j ∙h Q ~ β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))
+                  ( inv (eq-htpy (left-whisker-inv-htpy h J)))))
 ```
diff --git a/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md b/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
index 29f7575d70..cec7d05f1c 100644
--- a/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
+++ b/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
@@ -12,15 +12,14 @@ open import foundation.cartesian-morphisms-arrows
 open import foundation.commuting-triangles-of-maps
 open import foundation.commuting-triangles-of-morphisms-arrows
 open import foundation.cones-over-cospan-diagrams
+open import foundation.contractible-maps
 open import foundation.contractible-types
 open import foundation.coproducts-pullbacks
 open import foundation.dependent-pair-types
-open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.dependent-products-pullbacks
 open import foundation.dependent-sums-pullbacks
 open import foundation.diagonal-maps-cartesian-products-of-types
 open import foundation.equivalences
-open import foundation.homotopy-induction
 open import foundation.fibers-of-maps
 open import foundation.function-extensionality
 open import foundation.function-types
@@ -30,6 +29,7 @@ open import foundation.functoriality-dependent-pair-types
 open import foundation.functoriality-fibers-of-maps
 open import foundation.homotopies
 open import foundation.homotopies-morphisms-arrows
+open import foundation.homotopy-induction
 open import foundation.identity-types
 open import foundation.lifts-morphisms-arrows
 open import foundation.morphisms-arrows
@@ -40,6 +40,7 @@ open import foundation.pullbacks
 open import foundation.standard-pullbacks
 open import foundation.torsorial-type-families
 open import foundation.transport-along-identifications
+open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.unit-type
 open import foundation.universe-levels
 open import foundation.whiskering-homotopies-composition
@@ -55,22 +56,12 @@ open import orthogonal-factorization-systems.lifts-maps
 
 ## Idea
 
-A [morphism of arrows](foundation.morphisms-arrows.md) `β : g ⇒ h`,
-
-```text
-         β₀
-    B ------> C
-    |         |
-  g |    H    | h
-    ∨         ∨
-    B' -----> C',
-         β₁
-```
-
-is said to satisfy the
+A [morphism of arrows](foundation.morphisms-arrows.md) `β : g ⇒ h`, is said to
+satisfy the
 {{#concept "universal property" Disambiguation="cartesian morphisms of arrows"}}
-of cartesian morphisms of arrows if the natural map that assigns to every
-extension diagram of the form
+of cartesian morphisms of arrows if the natural map that assigns to, given
+another morphism of arrows `α : f ⇒ h`, and a
+[lifting diagram](foundation.lifts-morphisms-arrows.md) of the form
 
 ```text
           B
@@ -81,371 +72,184 @@ extension diagram of the form
     |     |     |
     |     ∨  H  |
   f |     B'    | h
-    |   ∧   \β₁ |
+    |   ∧   \   |
     |  /     \  |
     ∨ /       ∨ ∨
-    A' -------> C'
+    A' -------> C',
 ```
 
-the underlying data
+the underlying [lift](orthogonal-factorization-systems.lifts-maps.md) on the
+codomain
 
 ```text
-    A --------> C
-    |           |
-    |        H  |
-  f |     B'    | h
-    |   ∧   \β₁ |
-    | i/     \  |
-    ∨ /       ∨ ∨
+          B'
+        ∧   \
+     i /     \ β₁
+      /       ∨
     A' -------> C'
+          α₁
 ```
 
-is an equivalence.
+is an [equivalence](foundation-core.equivalences.md).
 
 ## Definitions
 
-```agda
-lift-codomain-lift-hom-arrow :
-  {l1 l2 l3 l4 l5 l6 : Level}
-  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
-  (f : A → A') (g : B → B') (h : C → C')
-  (α : hom-arrow f h) (β : hom-arrow g h) →
-  lift-hom-arrow f g h α β →
-  lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α)
-lift-codomain-lift-hom-arrow f g h α β γ =
-  ( map-codomain-lift-hom-arrow f g h α β γ ,
-    coh-codomain-lift-hom-arrow f g h α β γ)
-```
-
 ### The universal property of cartesian morphisms of arrows
 
 ```agda
-universal-property-cartesian-hom-arrow-Level :
-  (l1 l2 : Level) {l3 l4 l5 l6 : Level}
+module _
+  {l3 l4 l5 l6 : Level}
   {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
-  (g : B → B') (h : C → C') (β : hom-arrow g h) →
-  UU (lsuc l1 ⊔ lsuc l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6)
-universal-property-cartesian-hom-arrow-Level l1 l2 g h β =
-  {A : UU l1} {A' : UU l2} (f : A → A') (α : hom-arrow f h) →
-  is-equiv (lift-codomain-lift-hom-arrow f g h α β)
+  (g : B → B') (h : C → C') (β : hom-arrow g h)
+  where
+
+  universal-property-cartesian-hom-arrow-Level :
+    (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6)
+  universal-property-cartesian-hom-arrow-Level l1 l2 =
+    {A : UU l1} {A' : UU l2} (f : A → A') (α : hom-arrow f h) →
+    is-equiv (lift-codomain-lift-hom-arrow f g h α β)
+
+  universal-property-cartesian-hom-arrow : UUω
+  universal-property-cartesian-hom-arrow =
+    {l1 l2 : Level} → universal-property-cartesian-hom-arrow-Level l1 l2
+```
 
-universal-property-cartesian-hom-arrow :
+### The property of having unique lifts of lifts of the codomain
+
+```agda
+module _
   {l3 l4 l5 l6 : Level}
   {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
-  (g : B → B') (h : C → C') (β : hom-arrow g h) →
-  UUω
-universal-property-cartesian-hom-arrow g h β =
-  {l1 l2 : Level} → universal-property-cartesian-hom-arrow-Level l1 l2 g h β
+  (g : B → B') (h : C → C') (β : hom-arrow g h)
+  where
+
+  has-unique-lifts-hom-arrow-Level :
+    (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6)
+  has-unique-lifts-hom-arrow-Level l1 l2 =
+    {A : UU l1} {A' : UU l2} (f : A → A') (α : hom-arrow f h) →
+    (δ : lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α)) →
+    is-contr (lift-hom-arrow-of-lift-codomain-hom-arrow f g h α β δ)
+
+  has-unique-lifts-hom-arrow : UUω
+  has-unique-lifts-hom-arrow =
+    {l1 l2 : Level} → has-unique-lifts-hom-arrow-Level l1 l2
 ```
 
 ## Properties
 
-### Computing the fibers of `lift-codomain-lift-hom-arrow`
+### A morphism of arrows satisfies the universal property of cartesian morphisms if and only if it has unique lifts
 
 ```agda
 module _
-  {l1 l2 l3 l4 l5 l6 : Level}
-  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
-  (f : A → A') (g : B → B') (h : C → C')
-  (α : hom-arrow f h) (β : hom-arrow g h)
-  ( let β₁ = map-codomain-hom-arrow g h β)
+  {l3 l4 l5 l6 : Level}
+  {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (g : B → B') (h : C → C') (β : hom-arrow g h)
   where
 
-  is-lift-hom-arrow-of-lift-codomain-hom-arrow :
-    lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α) →
-    (A → B) → UU (l1 ⊔ l4 ⊔ l5 ⊔ l6)
-  is-lift-hom-arrow-of-lift-codomain-hom-arrow (i , I) j =
-    Σ ( coherence-hom-arrow f g j i)
-      ( λ H →
-        Σ ( coherence-triangle-maps
-            ( map-domain-hom-arrow f h α)
-            ( map-domain-hom-arrow g h β)
-            ( j))
-          ( λ J →
-            coherence-htpy-hom-arrow f h α
-              ( comp-hom-arrow f g h β (j , i , H))
-              ( J)
-              ( I)))
-
-  lift-hom-arrow-of-lift-codomain-hom-arrow :
-    lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α) →
-    UU (l1 ⊔ l3 ⊔ l4 ⊔ l5 ⊔ l6)
-  lift-hom-arrow-of-lift-codomain-hom-arrow i =
-    Σ (A → B) (is-lift-hom-arrow-of-lift-codomain-hom-arrow i)
-
-  compute-fiber-lift-codomain-lift-hom-arrow :
-    (δ : lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α)) →
-    fiber (lift-codomain-lift-hom-arrow f g h α β) δ ≃
-    lift-hom-arrow-of-lift-codomain-hom-arrow δ
-  compute-fiber-lift-codomain-lift-hom-arrow (δ , Hδ) =
-    equivalence-reasoning
-    Σ ( lift-hom-arrow f g h α β)
-      ( λ γ → lift-codomain-lift-hom-arrow f g h α β γ ＝ (δ , Hδ))
-    ≃ Σ ( lift-hom-arrow f g h α β)
-        ( λ γ →
-          Σ ( map-codomain-lift-hom-arrow f g h α β γ ~ δ)
-            ( λ Hi → coh-codomain-lift-hom-arrow f g h α β γ ∙h β₁ ·l Hi ~ Hδ))
-      by
-        equiv-tot
-          ( λ γ →
-            extensionality-lift
-              ( map-codomain-hom-arrow g h β)
-              ( map-codomain-hom-arrow f h α)
-              ( lift-codomain-lift-hom-arrow f g h α β γ)
-              ( δ , Hδ))
-    ≃ Σ ( Σ (A' → B') (_~ δ))
-        ( λ (i , Hi) →
-          Σ ( Σ ( coherence-triangle-maps
-                  ( map-codomain-hom-arrow f h α)
-                  ( map-codomain-hom-arrow g h β)
-                  ( i))
-                ( λ I → I ∙h β₁ ·l Hi ~ Hδ))
-            ( λ (I , HI) → lift-hom-arrow-of-lift-codomain-hom-arrow (i , I)))
-      by reassociate
-    ≃ Σ ( Σ ( coherence-triangle-maps
-              ( map-codomain-hom-arrow f h α)
-              ( map-codomain-hom-arrow g h β)
-              ( δ))
-            ( λ I → I ∙h β₁ ·l refl-htpy ~ Hδ))
-        ( λ (I , HI) → lift-hom-arrow-of-lift-codomain-hom-arrow (δ , I))
-      by left-unit-law-Σ-is-contr (is-torsorial-htpy' δ) (δ , refl-htpy)
-    ≃ Σ ( Σ ( coherence-triangle-maps
-              ( map-codomain-hom-arrow f h α)
-              ( map-codomain-hom-arrow g h β)
-              ( δ))
-            ( _~ Hδ))
-        ( λ (I , HI) → lift-hom-arrow-of-lift-codomain-hom-arrow (δ , I))
-      by
-        equiv-Σ-equiv-base
-          ( λ (I , HI) → lift-hom-arrow-of-lift-codomain-hom-arrow (δ , I))
-          ( equiv-tot (λ I → equiv-concat-htpy inv-htpy-right-unit-htpy Hδ))
-    ≃ lift-hom-arrow-of-lift-codomain-hom-arrow (δ , Hδ)
-      by left-unit-law-Σ-is-contr (is-torsorial-htpy' Hδ) (Hδ , refl-htpy)
-    where
-      reassociate :
-        Σ ( lift-hom-arrow f g h α β)
-        ( λ γ →
-          Σ ( map-codomain-lift-hom-arrow f g h α β γ ~ δ)
-            ( λ Hi →
-              coh-codomain-lift-hom-arrow f g h α β γ ∙h β₁ ·l Hi ~ Hδ)) ≃
-        Σ ( Σ (A' → B') (_~ δ))
-          ( λ (i , Hi) →
-            Σ ( Σ ( coherence-triangle-maps
-                    ( map-codomain-hom-arrow f h α)
-                    ( map-codomain-hom-arrow g h β)
-                    ( i))
-                  ( λ I → I ∙h β₁ ·l Hi ~ Hδ))
-                ( λ (I , HI) →
-                  lift-hom-arrow-of-lift-codomain-hom-arrow (i , I)))
-      reassociate =
-        ( ( λ (((γ₀ , γ₁ , Hγ) , (Γ₀ , Γ₁ , HΓ)) , (Hi , HI)) →
-            ( (γ₁ , Hi) , (Γ₁ , HI) , γ₀ , Hγ , Γ₀ , HΓ)) ,
-          ( is-equiv-is-invertible
-            ( λ ((γ₁ , Hi) , (Γ₁ , HI) , γ₀ , Hγ , Γ₀ , HΓ) →
-              ((γ₀ , γ₁ , Hγ) , (Γ₀ , Γ₁ , HΓ)) , (Hi , HI))
-            ( refl-htpy)
-            ( refl-htpy)))
-```
+  abstract
+    has-unique-lifts-up-cartesian-hom-arrow :
+      universal-property-cartesian-hom-arrow g h β →
+      has-unique-lifts-hom-arrow g h β
+    has-unique-lifts-up-cartesian-hom-arrow up-β f α δ =
+      is-contr-equiv' _
+        ( compute-fiber-lift-codomain-lift-hom-arrow f g h α β δ)
+        ( is-contr-map-is-equiv (up-β f α) δ)
 
-### Uniqueness of lifts of cartesian morphisms
+  abstract
+    up-cartesian-has-unique-lifts-hom-arrow :
+      has-unique-lifts-hom-arrow g h β →
+      universal-property-cartesian-hom-arrow g h β
+    up-cartesian-has-unique-lifts-hom-arrow L f α =
+      is-equiv-is-contr-map
+        ( λ δ →
+          is-contr-equiv _
+            ( compute-fiber-lift-codomain-lift-hom-arrow f g h α β δ)
+            ( L f α δ))
+```
 
-Given a cartesian morphism of arrows, then lifts along lifts in the codomain are
-unique.
+### A morphism of arrows has unique lifts if and only if it is cartesian
 
 ```agda
 module _
-  {l1 l2 l3 l4 l5 l6 : Level}
-  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
-  (f : A → A') (g : B → B') (h : C → C')
-  (α : hom-arrow f h) (β : hom-arrow g h)
-  (let β₁ = map-codomain-hom-arrow g h β)
-  (let Hβ = coh-hom-arrow g h β)
-  (let Hα = coh-hom-arrow f h α)
-  ((i , I) : lift (map-codomain-hom-arrow g h β) (map-codomain-hom-arrow f h α))
+  {l3 l4 l5 l6 : Level}
+  {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (g : B → B') (h : C → C') (β : hom-arrow g h)
   where
 
-  equiv-htpy-cone-is-lift-hom-arrow-of-lift-codomain-hom-arrow :
-    (j : A → B) →
-    is-lift-hom-arrow-of-lift-codomain-hom-arrow f g h α β (i , I) j ≃
-    htpy-cone β₁ h
-      ( g ∘ j , map-domain-hom-arrow g h β ∘ j , Hβ ·r j)
-      ( i ∘ f , map-domain-hom-arrow f h α , inv-htpy I ·r f ∙h Hα)
-  equiv-htpy-cone-is-lift-hom-arrow-of-lift-codomain-hom-arrow j =
-    equiv-Σ _
-      ( equiv-inv-htpy (i ∘ f) (g ∘ j))
-      ( λ H →
-        equiv-Σ _
-          ( equiv-inv-htpy
-            ( map-domain-hom-arrow f h α)
-            ( map-domain-hom-arrow g h β ∘ j))
-          ( λ J →
-            equivalence-reasoning
-            ( Hα ∙h h ·l J ~ I ·r f ∙h (β₁ ·l H ∙h Hβ ·r j))
-            ≃ ( I ·r f ∙h (β₁ ·l H ∙h Hβ ·r j) ~ Hα ∙h h ·l J)
-              by equiv-inv-htpy (Hα ∙h h ·l J) (I ·r f ∙h (β₁ ·l H ∙h Hβ ·r j))
-            ≃ ( β₁ ·l H ∙h Hβ ·r j ~ inv-htpy I ·r f ∙h (Hα ∙h h ·l J))
-              by
-                equiv-left-transpose-htpy-concat
-                  ( I ·r f)
-                  ( β₁ ·l H ∙h Hβ ·r j)
-                  ( Hα ∙h h ·l J)
-            ≃ ( Hβ ·r j ~
-                inv-htpy (β₁ ·l H) ∙h (inv-htpy I ·r f ∙h (Hα ∙h h ·l J)))
-              by
-                equiv-left-transpose-htpy-concat
-                  ( β₁ ·l H)
-                  ( Hβ ·r j)
-                  ( inv-htpy I ·r f ∙h (Hα ∙h h ·l J))
-            ≃ ( Hβ ·r j ~
-                β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h (Hα ∙h h ·l J)))
-              by
-                equiv-tr
-                  ( λ Q → Hβ ·r j ~ Q ∙h (inv-htpy I ·r f ∙h (Hα ∙h h ·l J)))
-                  ( inv (eq-htpy (left-whisker-inv-htpy β₁ H)))
-            ≃ ( Hβ ·r j ~
-                ( β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα)) ∙h h ·l J)
-              by
-                equiv-tr
-                  ( Hβ ·r j ~_)
-                  ( ( ap
-                      ( β₁ ·l (inv-htpy H) ∙h_)
-                      ( eq-htpy
-                        ( inv-htpy-assoc-htpy (inv-htpy I ·r f) Hα (h ·l J)))) ∙
-                    ( eq-htpy
-                      ( inv-htpy-assoc-htpy
-                        ( β₁ ·l (inv-htpy H))
-                        ( inv-htpy I ·r f ∙h Hα)
-                        ( h ·l J))))
-            ≃ ( Hβ ·r j ∙h inv-htpy (h ·l J) ~
-                β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))
-              by
-                equiv-right-transpose-htpy-concat'
-                  ( Hβ ·r j)
-                  ( β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))
-                  ( h ·l J)
-            ≃ ( Hβ ·r j ∙h h ·l inv-htpy J ~
-                β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))
-              by
-                equiv-tr
-                  ( λ Q →
-                    Hβ ·r j ∙h Q ~ β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))
-                  ( inv (eq-htpy (left-whisker-inv-htpy h J)))))
-
   abstract
-    uniqueness-lift-map-codomain-is-cartesian-hom-arrow :
+    has-unique-lifts-is-cartesian-hom-arrow :
       is-cartesian-hom-arrow g h β →
-      is-torsorial
-        ( is-lift-hom-arrow-of-lift-codomain-hom-arrow f g h α β (i , I))
-    uniqueness-lift-map-codomain-is-cartesian-hom-arrow H =
+      has-unique-lifts-hom-arrow g h β
+    has-unique-lifts-is-cartesian-hom-arrow H {A = A} f α (i , I) =
       is-contr-equiv _
         ( equiv-tot
-          ( equiv-htpy-cone-is-lift-hom-arrow-of-lift-codomain-hom-arrow))
-        ( uniqueness-universal-property-pullback β₁ h
+          ( equiv-htpy-cone-is-lift-hom-arrow-of-lift-codomain-hom-arrow
+            f g h α β (i , I)))
+        ( uniqueness-universal-property-pullback (map-codomain-hom-arrow g h β) h
           ( cone-hom-arrow g h β)
           ( up-pullback-cartesian-hom-arrow g h (β , H))
           ( A)
-          ( i ∘ f , map-domain-hom-arrow f h α , inv-htpy I ·r f ∙h Hα))
-```
+          ( i ∘ f ,
+            map-domain-hom-arrow f h α ,
+            inv-htpy I ·r f ∙h coh-hom-arrow f h α))
 
-<!-- This uniqueness proof also constructs an explicit lift, making the entire next section purposeless -->
+  abstract
+    is-cartesian-has-unique-lifts-hom-arrow :
+      has-unique-lifts-hom-arrow g h β →
+      is-cartesian-hom-arrow g h β
+    is-cartesian-has-unique-lifts-hom-arrow H =
+      is-pullback-universal-property-pullback
+        ( map-codomain-hom-arrow g h β)
+        ( h)
+        ( cone-hom-arrow g h β)
+        ( universal-property-pullback-uniqueness
+          ( map-codomain-hom-arrow g h β)
+          ( h)
+          ( cone-hom-arrow g h β)
+          ( λ A (f , i , I) →
+            ( is-contr-equiv' _
+              ( equiv-tot
+                ( equiv-htpy-cone-is-lift-hom-arrow-of-lift-codomain-hom-arrow
+                  ( f)
+                  ( g)
+                  ( h)
+                  ( i , map-codomain-hom-arrow g h β , I)
+                  ( β)
+                  ( id , refl-htpy)))
+              ( H f (i , map-codomain-hom-arrow g h β , I) (id , refl-htpy)))))
+```
 
-> What follows is another computation of the same. Should be deleted.
+### A morphism of arrows satisfies the universal property of cartesian morphisms if and only if it is cartesian
 
 ```agda
 module _
-  {l1 l2 l3 l4 l5 l6 : Level}
-  {A : UU l1} {A' : UU l2} {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
-  (f : A → A') (g : B → B') (h : C → C')
-  (β : cartesian-hom-arrow g h)
-  (α : hom-arrow f h)
-  (i : A' → B')
-  (H :
-    coherence-triangle-maps'
-      ( map-codomain-hom-arrow f h α)
-      ( map-codomain-cartesian-hom-arrow g h β)
-      ( i))
+  {l3 l4 l5 l6 : Level}
+  {B : UU l3} {B' : UU l4} {C : UU l5} {C' : UU l6}
+  (g : B → B') (h : C → C') (β : hom-arrow g h)
   where
 
   abstract
-    uniqueness-lift-map-codomain-cartesian-hom-arrow' :
-      is-torsorial
-        ( λ (j : A → B) →
-          Σ ( coherence-hom-arrow f g j i)
-            ( λ J →
-              Σ ( coherence-triangle-maps'
-                  ( map-domain-hom-arrow f h α)
-                  ( map-domain-cartesian-hom-arrow g h β)
-                  ( j))
-                ( λ K →
-                  coherence-htpy-hom-arrow f h
-                    ( comp-hom-arrow f g h
-                      ( hom-arrow-cartesian-hom-arrow g h β)
-                      ( j , i , J))
-                    ( α)
-                    ( K)
-                    ( H))))
-    uniqueness-lift-map-codomain-cartesian-hom-arrow' =
-      is-contr-equiv _
-        ( equiv-tot
-          ( λ j →
-            equiv-Σ _
-              ( equiv-inv-htpy (i ∘ f) (g ∘ j))
-              ( λ J →
-                equiv-tot
-                  ( λ K →
-                    equivalence-reasoning
-                    ( ( map-codomain-cartesian-hom-arrow g h β ·l J) ∙h
-                      ( coh-cartesian-hom-arrow g h β ·r j) ∙h
-                      ( h ·l K) ~
-                      ( H ·r f) ∙h
-                      ( coh-hom-arrow f h α))
-                    ≃ ( ( map-codomain-cartesian-hom-arrow g h β ·l J) ∙h
-                        ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h
-                          ( h ·l K)) ~
-                        ( H ·r f) ∙h
-                        ( coh-hom-arrow f h α))
-                      by
-                        equiv-tr
-                          ( λ Q → Q ~ ( H ·r f) ∙h ( coh-hom-arrow f h α))
-                          ( eq-htpy
-                            ( assoc-htpy
-                              ( map-codomain-cartesian-hom-arrow g h β ·l J)
-                              ( coh-cartesian-hom-arrow g h β ·r j)
-                              ( h ·l K)))
-                    ≃ ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K) ~
-                        ( inv-htpy
-                          ( map-codomain-cartesian-hom-arrow g h β ·l J)) ∙h
-                        ( ( H ·r f) ∙h (coh-hom-arrow f h α)))
-                      by
-                        equiv-left-transpose-htpy-concat
-                          ( map-codomain-cartesian-hom-arrow g h β ·l J)
-                          ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K))
-                          ( ( H ·r f) ∙h (coh-hom-arrow f h α))
-                    ≃ ( ( coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K) ~
-                        ( ( map-codomain-cartesian-hom-arrow g h β) ·l
-                          ( inv-htpy J)) ∙h
-                        ( ( H ·r f) ∙h (coh-hom-arrow f h α)))
-                      by
-                        equiv-tr
-                          ( λ Q →
-                            (coh-cartesian-hom-arrow g h β ·r j) ∙h (h ·l K) ~
-                            Q ∙h ((H ·r f) ∙h (coh-hom-arrow f h α)))
-                          ( inv
-                            ( eq-htpy
-                              ( left-whisker-inv-htpy
-                                ( map-codomain-cartesian-hom-arrow g h β)
-                                ( J))))))))
-        ( uniqueness-universal-property-pullback
-          ( map-codomain-cartesian-hom-arrow g h β)
-          ( h)
-          ( cone-cartesian-hom-arrow g h β)
-          ( up-pullback-cartesian-hom-arrow g h β)
-          ( A)
-          ( i ∘ f , pr1 α , H ·r f ∙h coh-hom-arrow f h α))
+    up-cartesian-is-cartesian-hom-arrow :
+      is-cartesian-hom-arrow g h β →
+      universal-property-cartesian-hom-arrow g h β
+    up-cartesian-is-cartesian-hom-arrow H =
+      up-cartesian-has-unique-lifts-hom-arrow g h β
+        ( has-unique-lifts-is-cartesian-hom-arrow g h β H)
+
+  abstract
+    is-cartesian-up-cartesian-hom-arrow :
+      universal-property-cartesian-hom-arrow g h β →
+      is-cartesian-hom-arrow g h β
+    is-cartesian-up-cartesian-hom-arrow H =
+      is-cartesian-has-unique-lifts-hom-arrow g h β
+        ( has-unique-lifts-up-cartesian-hom-arrow g h β H)
 ```
 
 ### Lifting cartesian morphisms along lifts of the codomain
 
+> The following computation is obsoleted by the formalizations above, but are
+> preserved for potential instructive value.
+
 Suppose given a cospan diagram of arrows
 
 ```text

From daeb23c484dde92fe8ad16b9be146192256657c8 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Tue, 16 Sep 2025 15:49:30 +0200
Subject: [PATCH 29/41] remove cringe `equiv-tr`s

---
 .../lifts-morphisms-arrows.lagda.md           | 39 +++++++++----------
 1 file changed, 18 insertions(+), 21 deletions(-)

diff --git a/src/foundation/lifts-morphisms-arrows.lagda.md b/src/foundation/lifts-morphisms-arrows.lagda.md
index ac8487736a..372add806c 100644
--- a/src/foundation/lifts-morphisms-arrows.lagda.md
+++ b/src/foundation/lifts-morphisms-arrows.lagda.md
@@ -26,6 +26,7 @@ open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 open import foundation.whiskering-homotopies-composition
+open import foundation.whiskering-homotopies-concatenation
 
 open import orthogonal-factorization-systems.lifts-maps
 ```
@@ -283,26 +284,21 @@ module _
                   ( β₁ ·l H)
                   ( Hβ ·r j)
                   ( inv-htpy I ·r f ∙h (Hα ∙h h ·l J))
-            ≃ ( Hβ ·r j ~
-                β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h (Hα ∙h h ·l J)))
-              by
-                equiv-tr
-                  ( λ Q → Hβ ·r j ~ Q ∙h (inv-htpy I ·r f ∙h (Hα ∙h h ·l J)))
-                  ( inv (eq-htpy (left-whisker-inv-htpy β₁ H)))
             ≃ ( Hβ ·r j ~
                 ( β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα)) ∙h h ·l J)
               by
-                equiv-tr
-                  ( Hβ ·r j ~_)
-                  ( ( ap
-                      ( β₁ ·l (inv-htpy H) ∙h_)
-                      ( eq-htpy
-                        ( inv-htpy-assoc-htpy (inv-htpy I ·r f) Hα (h ·l J)))) ∙
-                    ( eq-htpy
-                      ( inv-htpy-assoc-htpy
-                        ( β₁ ·l (inv-htpy H))
-                        ( inv-htpy I ·r f ∙h Hα)
-                        ( h ·l J))))
+                equiv-concat-htpy'
+                  ( Hβ ·r j)
+                  ( ( right-whisker-concat-htpy
+                      ( inv-htpy (left-whisker-inv-htpy β₁ H))
+                      ( inv-htpy I ·r f ∙h (Hα ∙h h ·l J))) ∙h
+                    ( left-whisker-concat-htpy
+                      ( β₁ ·l (inv-htpy H))
+                      ( inv-htpy-assoc-htpy (inv-htpy I ·r f) Hα (h ·l J))) ∙h
+                    ( inv-htpy-assoc-htpy
+                      ( β₁ ·l (inv-htpy H))
+                      ( inv-htpy I ·r f ∙h Hα)
+                      ( h ·l J)))
             ≃ ( Hβ ·r j ∙h inv-htpy (h ·l J) ~
                 β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))
               by
@@ -313,8 +309,9 @@ module _
             ≃ ( Hβ ·r j ∙h h ·l inv-htpy J ~
                 β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))
               by
-                equiv-tr
-                  ( λ Q →
-                    Hβ ·r j ∙h Q ~ β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))
-                  ( inv (eq-htpy (left-whisker-inv-htpy h J)))))
+                equiv-concat-htpy
+                  ( left-whisker-concat-htpy
+                    ( Hβ ·r j)
+                    ( left-whisker-inv-htpy h J))
+                  ( β₁ ·l inv-htpy H ∙h (inv-htpy I ·r f ∙h Hα))))
 ```

From 8c7faaeadc78e10162a8041915b2940ec9014777 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Tue, 16 Sep 2025 15:52:04 +0200
Subject: [PATCH 30/41] edits

---
 .../universal-property-cartesian-morphisms-arrows.lagda.md  | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md b/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
index cec7d05f1c..3c9301ddd8 100644
--- a/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
+++ b/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
@@ -245,7 +245,7 @@ module _
         ( has-unique-lifts-up-cartesian-hom-arrow g h β H)
 ```
 
-### Lifting cartesian morphisms along lifts of the codomain
+### Explicit construction of lifts along cartesian morphisms
 
 > The following computation is obsoleted by the formalizations above, but are
 > preserved for potential instructive value.
@@ -281,7 +281,7 @@ if `α` is.
 pullback property of `β`. The rest is a reiteration of the 3-for-2 property of
 cartesian morphisms. ∎
 
-We begin by constructing the commuting triangle of morphisms of arrows:
+We construct the commuting triangle of morphisms of arrows:
 
 ```agda
 module _
@@ -416,7 +416,7 @@ module _
       ( inv-coherence-triangle-hom-arrow-lift-map-codomain-cartesian-hom-arrow)
 ```
 
-Now, if `α` was cartesian to begin with, the lift is also.
+If `α` was cartesian to begin with, the lift is also.
 
 ```agda
 module _

From c96f223ef2a433ec495319069a7fd09d721e140b Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Tue, 16 Sep 2025 17:00:15 +0200
Subject: [PATCH 31/41] delete some stuff

---
 .../cartesian-morphisms-arrows.lagda.md       |  5 +-
 ...ormations-polynomial-endofunctors.lagda.md | 56 +------------------
 2 files changed, 6 insertions(+), 55 deletions(-)

diff --git a/src/foundation/cartesian-morphisms-arrows.lagda.md b/src/foundation/cartesian-morphisms-arrows.lagda.md
index 288eadafba..98047147de 100644
--- a/src/foundation/cartesian-morphisms-arrows.lagda.md
+++ b/src/foundation/cartesian-morphisms-arrows.lagda.md
@@ -265,13 +265,13 @@ module _
   (f : A → B) (g : X → Y) (α : hom-arrow f g)
   where
 
-  is-cartesian-is-cartesian-transpose-cartesian-hom-arrow :
+  is-cartesian-is-cartesian-transpose-hom-arrow :
     is-cartesian-hom-arrow
       ( map-domain-hom-arrow f g α)
       ( map-codomain-hom-arrow f g α)
       ( transpose-hom-arrow f g α) →
     is-cartesian-hom-arrow f g α
-  is-cartesian-is-cartesian-transpose-cartesian-hom-arrow =
+  is-cartesian-is-cartesian-transpose-hom-arrow =
     is-pullback-swap-cone'
       ( map-codomain-hom-arrow f g α)
       ( g)
@@ -881,6 +881,7 @@ module _
 
 ## See also
 
+- [The universal property of cartesian morphisms of arrows](foundation.universal-property-cartesian-morphisms-arrows.md)
 - [Cocartesian morphisms of arrows](synthetic-homotopy-theory.cocartesian-morphisms-arrows.md)
   for the dual.
 - [Diagonals of morphisms of arrows](foundation.diagonals-of-morphisms-arrows.md)
diff --git a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
index 2302731798..3d6f1b34c8 100644
--- a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
@@ -11,12 +11,15 @@ open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
 open import foundation.binary-homotopies
 open import foundation.cartesian-morphisms-arrows
+open import foundation.homotopies-morphisms-arrows
 open import foundation.commuting-squares-of-homotopies
 open import foundation.commuting-squares-of-maps
 open import foundation.cones-over-cospan-diagrams
 open import foundation.contractible-types
+open import foundation.lifts-morphisms-arrows
 open import foundation.dependent-pair-types
 open import foundation.equality-dependent-function-types
+open import foundation.universal-property-cartesian-morphisms-arrows
 open import foundation.equivalences
 open import foundation.function-types
 open import foundation.functoriality-dependent-pair-types
@@ -298,59 +301,6 @@ $f : X → Y$, we have a morphism of arrows in the slice above $α_{*}$:
 and so by the right-cancellation property of cartesian squares the naturality
 square at $f$ is cartesian. ∎
 
-```agda
-module _
-  {l1 l2 l3 l4 l : Level}
-  (𝑃 : polynomial-endofunctor l1 l2)
-  (𝑄 : polynomial-endofunctor l3 l4)
-  (α : natural-transformation-polynomial-endofunctor l 𝑃 𝑄)
-  (let α₀ = type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α)
-  where
-
-  is-cartesian-at-terminal-map-is-cartesian-natural-transformation-polynomial-endofunctor :
-    is-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α →
-    is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor
-      𝑃 𝑄 α
-  is-cartesian-at-terminal-map-is-cartesian-natural-transformation-polynomial-endofunctor
-    H {X} =
-    H (raise-terminal-map X)
-
-  abstract
-    is-cartesian-is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor :
-      is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor
-        𝑃 𝑄 α →
-      is-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α
-    is-cartesian-is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor
-      H {X} {Y} f =
-      is-pullback-top-square-vertical-triangle
-        α₀
-        ( map-polynomial-endofunctor 𝑄 (raise-terminal-map Y))
-        ( map-polynomial-endofunctor 𝑄 f)
-        ( map-polynomial-endofunctor 𝑄 (raise-terminal-map X))
-        ( cone-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α
-          ( raise-terminal-map Y))
-        ( cone-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α
-          ( f))
-        ( cone-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α
-          ( raise-terminal-map X))
-        ( refl-htpy)
-        ( refl-htpy ,
-          refl-htpy ,
-          ( λ (a , h) →
-            equational-reasoning
-            pr2 α (raise-terminal-map Y) (a , f ∘ h) ∙ ap (tot (λ x₁ f₁ a → map-raise star)) (pr2 α f (a , h)) ∙ refl
-            ＝
-              pr2 α (raise-terminal-map Y) (a , f ∘ h) ∙
-              ap (λ z → pair (pr1 z) (λ z₁ → map-raise star)) (pr2 α f (a , h))
-              by right-unit
-            ＝ {!   !} by {!   !}
-            ＝ pr2 α (raise-terminal-map X) (a , h) by {!   !})) -- these are two equalities in the type (Σ 𝑄₀ , 1), so they only need to be equal on first components.
-        ( H {Y})
-        ( H {X})
-```
-
-<!-- TODO: use uniqueness of cartesian lifts -->
-
 This is mentioned as Remark 2.1.4 in {{#cite GHK22}}.
 
 ## References

From 0b1130da27f2cea7110b46731ef68f72d4dea440 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Tue, 16 Sep 2025 17:01:24 +0200
Subject: [PATCH 32/41] singluar `shape` and `position`

---
 ...morphisms-polynomial-endofunctors.lagda.md | 66 +++++++++----------
 ...morphisms-polynomial-endofunctors.lagda.md | 38 +++++------
 src/trees/polynomial-endofunctors.lagda.md    | 22 +++----
 ...univalent-polynomial-endofunctors.lagda.md |  4 +-
 4 files changed, 65 insertions(+), 65 deletions(-)

diff --git a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
index 61d0a18ff4..6e7652ad50 100644
--- a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
@@ -82,7 +82,7 @@ module _
 
   is-cartesian-hom-polynomial-endofunctor : UU (l1 ⊔ l2 ⊔ l4)
   is-cartesian-hom-polynomial-endofunctor =
-    (a : shapes-polynomial-endofunctor 𝑃) →
+    (a : shape-polynomial-endofunctor 𝑃) →
     is-equiv (positions-hom-polynomial-endofunctor 𝑃 𝑄 α a)
 
   is-prop-is-cartesian-hom-polynomial-endofunctor :
@@ -114,10 +114,10 @@ make-cartesian-hom-polynomial-endofunctor :
   {l1 l2 l3 l4 : Level}
   (𝑃 : polynomial-endofunctor l1 l2)
   (𝑄 : polynomial-endofunctor l3 l4)
-  (α₀ : shapes-polynomial-endofunctor 𝑃 → shapes-polynomial-endofunctor 𝑄) →
-  ( (a : shapes-polynomial-endofunctor 𝑃) →
-    positions-polynomial-endofunctor 𝑄 (α₀ a) ≃
-    positions-polynomial-endofunctor 𝑃 a) →
+  (α₀ : shape-polynomial-endofunctor 𝑃 → shape-polynomial-endofunctor 𝑄) →
+  ( (a : shape-polynomial-endofunctor 𝑃) →
+    position-polynomial-endofunctor 𝑄 (α₀ a) ≃
+    position-polynomial-endofunctor 𝑃 a) →
   cartesian-hom-polynomial-endofunctor 𝑃 𝑄
 make-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α₀ α₁ =
   ( ( α₀ , map-equiv ∘ α₁) , is-equiv-map-equiv ∘ α₁)
@@ -133,18 +133,18 @@ module _
   hom-cartesian-hom-polynomial-endofunctor = pr1 α
 
   shapes-cartesian-hom-polynomial-endofunctor :
-    shapes-polynomial-endofunctor 𝑃 → shapes-polynomial-endofunctor 𝑄
+    shape-polynomial-endofunctor 𝑃 → shape-polynomial-endofunctor 𝑄
   shapes-cartesian-hom-polynomial-endofunctor =
     shapes-hom-polynomial-endofunctor 𝑃 𝑄
       hom-cartesian-hom-polynomial-endofunctor
 
   positions-cartesian-hom-polynomial-endofunctor :
-    (a : shapes-polynomial-endofunctor 𝑃) →
-    positions-polynomial-endofunctor 𝑄
+    (a : shape-polynomial-endofunctor 𝑃) →
+    position-polynomial-endofunctor 𝑄
       ( shapes-hom-polynomial-endofunctor 𝑃 𝑄
         ( hom-cartesian-hom-polynomial-endofunctor)
         ( a)) →
-    positions-polynomial-endofunctor 𝑃 a
+    position-polynomial-endofunctor 𝑃 a
   positions-cartesian-hom-polynomial-endofunctor =
     positions-hom-polynomial-endofunctor 𝑃 𝑄
       hom-cartesian-hom-polynomial-endofunctor
@@ -162,12 +162,12 @@ module _
   is-cartesian-cartesian-hom-polynomial-endofunctor = pr2 α
 
   equiv-positions-cartesian-hom-polynomial-endofunctor :
-    (a : shapes-polynomial-endofunctor 𝑃) →
-    positions-polynomial-endofunctor 𝑄
+    (a : shape-polynomial-endofunctor 𝑃) →
+    position-polynomial-endofunctor 𝑄
       ( shapes-hom-polynomial-endofunctor 𝑃 𝑄
         ( hom-cartesian-hom-polynomial-endofunctor)
         ( a)) ≃
-    positions-polynomial-endofunctor 𝑃 a
+    position-polynomial-endofunctor 𝑃 a
   equiv-positions-cartesian-hom-polynomial-endofunctor a =
     ( positions-cartesian-hom-polynomial-endofunctor a ,
       is-cartesian-cartesian-hom-polynomial-endofunctor a)
@@ -247,11 +247,11 @@ module _
 
   cartesian-hom-polynomial-endofunctor' : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
   cartesian-hom-polynomial-endofunctor' =
-    Σ ( shapes-polynomial-endofunctor 𝑃 → shapes-polynomial-endofunctor 𝑄)
+    Σ ( shape-polynomial-endofunctor 𝑃 → shape-polynomial-endofunctor 𝑄)
       ( λ α₀ →
-        ((a : shapes-polynomial-endofunctor 𝑃) →
-          positions-polynomial-endofunctor 𝑄 (α₀ a) ≃
-          positions-polynomial-endofunctor 𝑃 a))
+        ((a : shape-polynomial-endofunctor 𝑃) →
+          position-polynomial-endofunctor 𝑄 (α₀ a) ≃
+          position-polynomial-endofunctor 𝑃 a))
 
   reassociate-type-cartesian-hom-polynomial-endofunctor :
     cartesian-hom-polynomial-endofunctor 𝑃 𝑄 ≃
@@ -274,9 +274,9 @@ module _
 
   is-trunc-succ-cartesian-hom-polynomial-endofunctor' :
     (k : 𝕋) →
-    is-trunc (succ-𝕋 k) (shapes-polynomial-endofunctor 𝑄) →
-    ( (c : shapes-polynomial-endofunctor 𝑄) →
-      is-trunc (succ-𝕋 k) (positions-polynomial-endofunctor 𝑄 c)) →
+    is-trunc (succ-𝕋 k) (shape-polynomial-endofunctor 𝑄) →
+    ( (c : shape-polynomial-endofunctor 𝑄) →
+      is-trunc (succ-𝕋 k) (position-polynomial-endofunctor 𝑄 c)) →
     is-trunc (succ-𝕋 k) (cartesian-hom-polynomial-endofunctor' 𝑃 𝑄)
   is-trunc-succ-cartesian-hom-polynomial-endofunctor' k hQ hQ' =
     is-trunc-Σ
@@ -288,9 +288,9 @@ module _
 
   is-trunc-succ-cartesian-hom-polynomial-endofunctor :
     (k : 𝕋) →
-    is-trunc (succ-𝕋 k) (shapes-polynomial-endofunctor 𝑄) →
-    ( (c : shapes-polynomial-endofunctor 𝑄) →
-      is-trunc (succ-𝕋 k) (positions-polynomial-endofunctor 𝑄 c)) →
+    is-trunc (succ-𝕋 k) (shape-polynomial-endofunctor 𝑄) →
+    ( (c : shape-polynomial-endofunctor 𝑄) →
+      is-trunc (succ-𝕋 k) (position-polynomial-endofunctor 𝑄 c)) →
     is-trunc (succ-𝕋 k) (cartesian-hom-polynomial-endofunctor 𝑃 𝑄)
   is-trunc-succ-cartesian-hom-polynomial-endofunctor k hQ hQ' =
     is-trunc-equiv
@@ -314,8 +314,8 @@ module _
   where
 
   compute-fiber-type-cartesian-hom-polynomial-endofunctor :
-    (c : shapes-polynomial-endofunctor 𝑄)
-    (x : positions-polynomial-endofunctor 𝑄 c → X) →
+    (c : shape-polynomial-endofunctor 𝑄)
+    (x : position-polynomial-endofunctor 𝑄 c → X) →
     fiber (type-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α {X = X}) (c , x) ≃
     fiber α₀ c
   compute-fiber-type-cartesian-hom-polynomial-endofunctor c x =
@@ -325,10 +325,10 @@ module _
           ( λ (a , p) →
             fiber
               ( precomp (α₁ a) X)
-              ( inv-tr (λ c → positions-polynomial-endofunctor 𝑄 c → X) p x))
+              ( inv-tr (λ c → position-polynomial-endofunctor 𝑄 c → X) p x))
         by
           compute-fiber-map-Σ
-            ( λ c → positions-polynomial-endofunctor 𝑄 c → X)
+            ( λ c → position-polynomial-endofunctor 𝑄 c → X)
             ( α₀)
             ( λ a → precomp (α₁ a) X)
             ( c , x)
@@ -341,7 +341,7 @@ module _
                   ( α₁ a)
                   ( is-cartesian-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α a)
                   ( X))
-                ( inv-tr (λ c → positions-polynomial-endofunctor 𝑄 c → X) p x))
+                ( inv-tr (λ c → position-polynomial-endofunctor 𝑄 c → X) p x))
 ```
 
 ### The associated natural transformation of a cartesian morphism is cartesian
@@ -388,23 +388,23 @@ module _
   (𝑃 : polynomial-endofunctor l1 l2)
   (𝑄 : polynomial-endofunctor l3 l4)
   (α : cartesian-natural-transformation-polynomial-endofunctor l2 𝑃 𝑄)
-  (let 𝑃₀ = shapes-polynomial-endofunctor 𝑃)
-  (let 𝑃₁ = positions-polynomial-endofunctor 𝑃)
+  (let 𝑃₀ = shape-polynomial-endofunctor 𝑃)
+  (let 𝑃₁ = position-polynomial-endofunctor 𝑃)
   (let α₀ = type-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α)
   where
 
   shapes-cartesian-natural-transformation-polynomial-endofunctor :
-    shapes-polynomial-endofunctor 𝑃 → shapes-polynomial-endofunctor 𝑄
+    shape-polynomial-endofunctor 𝑃 → shape-polynomial-endofunctor 𝑄
   shapes-cartesian-natural-transformation-polynomial-endofunctor =
     shapes-natural-transformation-polynomial-endofunctor 𝑃 𝑄
       ( natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
           𝑃 𝑄 α)
 
   positions-cartesian-natural-transformation-polynomial-endofunctor :
-    (a : shapes-polynomial-endofunctor 𝑃) →
-    positions-polynomial-endofunctor 𝑄
+    (a : shape-polynomial-endofunctor 𝑃) →
+    position-polynomial-endofunctor 𝑄
       ( shapes-cartesian-natural-transformation-polynomial-endofunctor a) →
-    positions-polynomial-endofunctor 𝑃 a
+    position-polynomial-endofunctor 𝑃 a
   positions-cartesian-natural-transformation-polynomial-endofunctor =
     positions-natural-transformation-polynomial-endofunctor 𝑃 𝑄
       ( natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
diff --git a/src/trees/morphisms-polynomial-endofunctors.lagda.md b/src/trees/morphisms-polynomial-endofunctors.lagda.md
index 56c62529c1..98242f0b78 100644
--- a/src/trees/morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/morphisms-polynomial-endofunctors.lagda.md
@@ -73,13 +73,13 @@ module _
   where
 
   shapes-hom-polynomial-endofunctor :
-    shapes-polynomial-endofunctor 𝑃 → shapes-polynomial-endofunctor 𝑄
+    shape-polynomial-endofunctor 𝑃 → shape-polynomial-endofunctor 𝑄
   shapes-hom-polynomial-endofunctor = pr1 α
 
   positions-hom-polynomial-endofunctor :
-    (a : shapes-polynomial-endofunctor 𝑃) →
-    positions-polynomial-endofunctor 𝑄 (shapes-hom-polynomial-endofunctor a) →
-    positions-polynomial-endofunctor 𝑃 a
+    (a : shape-polynomial-endofunctor 𝑃) →
+    position-polynomial-endofunctor 𝑄 (shapes-hom-polynomial-endofunctor a) →
+    position-polynomial-endofunctor 𝑃 a
   positions-hom-polynomial-endofunctor = pr2 α
 
   type-hom-polynomial-endofunctor :
@@ -88,7 +88,7 @@ module _
     type-polynomial-endofunctor 𝑄 X
   type-hom-polynomial-endofunctor {X = X} =
     map-Σ
-      ( λ c → positions-polynomial-endofunctor 𝑄 c → X)
+      ( λ c → position-polynomial-endofunctor 𝑄 c → X)
       ( shapes-hom-polynomial-endofunctor)
       ( λ a → precomp (positions-hom-polynomial-endofunctor a) X)
 ```
@@ -135,13 +135,13 @@ module _
     Σ ( shapes-hom-polynomial-endofunctor 𝑃 𝑄 α ~
         shapes-hom-polynomial-endofunctor 𝑃 𝑄 β)
       ( λ H →
-        (a : shapes-polynomial-endofunctor 𝑃)
+        (a : shape-polynomial-endofunctor 𝑃)
         (d :
-          positions-polynomial-endofunctor 𝑄
+          position-polynomial-endofunctor 𝑄
             ( shapes-hom-polynomial-endofunctor 𝑃 𝑄 α a)) →
         positions-hom-polynomial-endofunctor 𝑃 𝑄 α a d ＝
         positions-hom-polynomial-endofunctor 𝑃 𝑄 β a
-          ( tr (positions-polynomial-endofunctor 𝑄) (H a) d))
+          ( tr (position-polynomial-endofunctor 𝑄) (H a) d))
 
   refl-htpy-hom-polynomial-endofunctor :
     (α : hom-polynomial-endofunctor 𝑃 𝑄) → htpy-hom-polynomial-endofunctor α α
@@ -199,9 +199,9 @@ module _
 
   is-trunc-hom-polynomial-endofunctor :
     (k : 𝕋) →
-    is-trunc k (shapes-polynomial-endofunctor 𝑄) →
-    ( (a : shapes-polynomial-endofunctor 𝑃) →
-      is-trunc k (positions-polynomial-endofunctor 𝑃 a)) →
+    is-trunc k (shape-polynomial-endofunctor 𝑄) →
+    ( (a : shape-polynomial-endofunctor 𝑃) →
+      is-trunc k (position-polynomial-endofunctor 𝑃 a)) →
     is-trunc k (hom-polynomial-endofunctor 𝑃 𝑄)
   is-trunc-hom-polynomial-endofunctor k hQ hP =
     is-trunc-Σ
@@ -266,15 +266,15 @@ module _
   where
 
   shapes-natural-transformation-polynomial-endofunctor :
-    shapes-polynomial-endofunctor 𝑃 → shapes-polynomial-endofunctor 𝑄
+    shape-polynomial-endofunctor 𝑃 → shape-polynomial-endofunctor 𝑄
   shapes-natural-transformation-polynomial-endofunctor a =
     pr1 (type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α (a , id))
 
   positions-natural-transformation-polynomial-endofunctor :
-    (a : shapes-polynomial-endofunctor 𝑃) →
-    positions-polynomial-endofunctor 𝑄
+    (a : shape-polynomial-endofunctor 𝑃) →
+    position-polynomial-endofunctor 𝑄
       ( shapes-natural-transformation-polynomial-endofunctor a) →
-    positions-polynomial-endofunctor 𝑃 a
+    position-polynomial-endofunctor 𝑃 a
   positions-natural-transformation-polynomial-endofunctor a =
     pr2 (type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α (a , id))
 
@@ -293,9 +293,9 @@ module _
   (𝑃 : polynomial-endofunctor l1 l2)
   (𝑄 : polynomial-endofunctor l3 l4)
   (α@(α₀ , α₁) : hom-polynomial-endofunctor 𝑃 𝑄)
-  (let 𝑃₁ = positions-polynomial-endofunctor 𝑃)
-  (let 𝑄₀ = shapes-polynomial-endofunctor 𝑄)
-  (let 𝑄₁ = positions-polynomial-endofunctor 𝑄)
+  (let 𝑃₁ = position-polynomial-endofunctor 𝑃)
+  (let 𝑄₀ = shape-polynomial-endofunctor 𝑄)
+  (let 𝑄₁ = position-polynomial-endofunctor 𝑄)
   {X : UU l5}
   where
 
@@ -320,7 +320,7 @@ module _
               ( inv-tr (λ c' → pr2 𝑄 c' → X) p x))
         by
           compute-fiber-map-Σ
-            ( λ c → positions-polynomial-endofunctor 𝑄 c → X)
+            ( λ c → position-polynomial-endofunctor 𝑄 c → X)
             ( α₀)
             ( λ a → precomp (α₁ a) X)
             ( c , x)
diff --git a/src/trees/polynomial-endofunctors.lagda.md b/src/trees/polynomial-endofunctors.lagda.md
index 044cec1a61..de31ed9f8e 100644
--- a/src/trees/polynomial-endofunctors.lagda.md
+++ b/src/trees/polynomial-endofunctors.lagda.md
@@ -57,11 +57,11 @@ module _
   {l1 l2 : Level} (𝑃 : polynomial-endofunctor l1 l2)
   where
 
-  shapes-polynomial-endofunctor : UU l1
-  shapes-polynomial-endofunctor = pr1 𝑃
+  shape-polynomial-endofunctor : UU l1
+  shape-polynomial-endofunctor = pr1 𝑃
 
-  positions-polynomial-endofunctor : shapes-polynomial-endofunctor → UU l2
-  positions-polynomial-endofunctor = pr2 𝑃
+  position-polynomial-endofunctor : shape-polynomial-endofunctor → UU l2
+  position-polynomial-endofunctor = pr2 𝑃
 
 make-polynomial-endofunctor :
   {l1 l2 : Level} {A : UU l1} → (A → UU l2) → polynomial-endofunctor l1 l2
@@ -206,20 +206,20 @@ module _
   where
 
   compute-fiber-map-polynomial-endofunctor :
-    (a : shapes-polynomial-endofunctor 𝑃)
-    (y : positions-polynomial-endofunctor 𝑃 a → Y) →
+    (a : shape-polynomial-endofunctor 𝑃)
+    (y : position-polynomial-endofunctor 𝑃 a → Y) →
     fiber (map-polynomial-endofunctor 𝑃 f) (a , y) ≃
-    ( (b : positions-polynomial-endofunctor 𝑃 a) → fiber f (y b))
+    ( (b : position-polynomial-endofunctor 𝑃 a) → fiber f (y b))
   compute-fiber-map-polynomial-endofunctor a y =
     equivalence-reasoning
     fiber (map-polynomial-endofunctor 𝑃 f) (a , y)
-    ≃ fiber (postcomp (positions-polynomial-endofunctor 𝑃 a) f) y
+    ≃ fiber (postcomp (position-polynomial-endofunctor 𝑃 a) f) y
       by
         compute-fiber-tot
-          ( λ a → postcomp (positions-polynomial-endofunctor 𝑃 a) f)
+          ( λ a → postcomp (position-polynomial-endofunctor 𝑃 a) f)
           ( a , y)
-    ≃ ((b : positions-polynomial-endofunctor 𝑃 a) → fiber f (y b))
-      by inv-compute-Π-fiber-postcomp (positions-polynomial-endofunctor 𝑃 a) f y
+    ≃ ((b : position-polynomial-endofunctor 𝑃 a) → fiber f (y b))
+      by inv-compute-Π-fiber-postcomp (position-polynomial-endofunctor 𝑃 a) f y
 ```
 
 ## See also
diff --git a/src/trees/univalent-polynomial-endofunctors.lagda.md b/src/trees/univalent-polynomial-endofunctors.lagda.md
index aac0e4c1d4..62a9e61f01 100644
--- a/src/trees/univalent-polynomial-endofunctors.lagda.md
+++ b/src/trees/univalent-polynomial-endofunctors.lagda.md
@@ -90,13 +90,13 @@ module _
 
   shapes-univalent-polynomial-endofunctor : UU l1
   shapes-univalent-polynomial-endofunctor =
-    shapes-polynomial-endofunctor
+    shape-polynomial-endofunctor
       polynomial-endofunctor-univalent-polynomial-endofunctor
 
   positions-univalent-polynomial-endofunctor :
     shapes-univalent-polynomial-endofunctor → UU l2
   positions-univalent-polynomial-endofunctor =
-    positions-polynomial-endofunctor
+    position-polynomial-endofunctor
       polynomial-endofunctor-univalent-polynomial-endofunctor
 
   is-univalent-univalent-polynomial-endofunctor :

From 1c9f007657c1740cd366b5e2e47e463f3e405a69 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Tue, 16 Sep 2025 22:40:17 +0200
Subject: [PATCH 33/41] casing

---
 references.bib | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/references.bib b/references.bib
index a906360dac..8eb952ef03 100644
--- a/references.bib
+++ b/references.bib
@@ -522,7 +522,7 @@ @article{GHK22
 
 @article{GK12,
   title         = {Polynomial functors and polynomial monads},
-  author        = {GAMBINO, NICOLA and KOCK, JOACHIM},
+  author        = {Gambino, Nicola and Kock, Joachim},
   year          = 2012,
   month         = sep,
   journal       = {Mathematical Proceedings of the Cambridge Philosophical Society},

From 19ef9008e8b2a6b7210f58b14f5dbf59c15e2ff7 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Fri, 19 Sep 2025 07:48:07 +0200
Subject: [PATCH 34/41] random edits

---
 .../double-negation-images.lagda.md           |  8 ++---
 src/foundation/noninjective-maps.lagda.md     | 34 +++++++++----------
 2 files changed, 21 insertions(+), 21 deletions(-)

diff --git a/src/foundation/double-negation-images.lagda.md b/src/foundation/double-negation-images.lagda.md
index 7fdda611c5..c74046f3c0 100644
--- a/src/foundation/double-negation-images.lagda.md
+++ b/src/foundation/double-negation-images.lagda.md
@@ -61,7 +61,7 @@ such that `l` is double negation dense, i.e., its
 [fibers](foundation-core.fibers-of-maps.md) are
 [nonempty](foundation.double-negation.md), and `r` is a double negation stable
 embedding, i.e., its fibers are [propositions](foundation-core.propositions.md)
-and satisfy the property that if they are nonempty then they are inhabited.
+that satisfy the property that if they are nonempty then they are inhabited.
 
 This factorization is unique in the sense that any other commuting triangle
 
@@ -79,9 +79,9 @@ modality, which is shown in
 [`foundation.double-negation-modality`](foundation.double-negation-modality.md).
 
 The double negation image factorization is in one sense an approximation to the
-[image factorization](foundation.images.md) which satisfies a restricted
-universal property that only applies to
-[negative statements](foundation-core.negation.md) and does not rely on the
+[image factorization](foundation.images.md) that satisfies a restricted
+universal property which only applies to
+[negative statements](foundation-core.negation.md), and does not rely on the
 existence of
 [propositional truncations](foundation.propositional-truncations.md). On the
 other hand, the double negation image inclusion `r` satisfies the additional
diff --git a/src/foundation/noninjective-maps.lagda.md b/src/foundation/noninjective-maps.lagda.md
index b5a08aeb11..dc8672eef6 100644
--- a/src/foundation/noninjective-maps.lagda.md
+++ b/src/foundation/noninjective-maps.lagda.md
@@ -41,23 +41,6 @@ if there [exists](foundation.existential-quantification.md) a
 
 ## Definitions
 
-### Not injective maps
-
-```agda
-module _
-  {l1 l2 : Level} {A : UU l1} {B : UU l2}
-  where
-
-  is-not-injective : (A → B) → UU (l1 ⊔ l2)
-  is-not-injective f = ¬ (is-injective f)
-
-  is-prop-is-not-injective : {f : A → B} → is-prop (is-not-injective f)
-  is-prop-is-not-injective = is-prop-neg
-
-  is-not-injective-Prop : (A → B) → Prop (l1 ⊔ l2)
-  is-not-injective-Prop f = (is-not-injective f , is-prop-is-not-injective)
-```
-
 ### Noninjective maps
 
 ```agda
@@ -92,6 +75,23 @@ module _
   is-noninjective-map-noninjective-map = pr2 f
 ```
 
+### Not injective maps
+
+```agda
+module _
+  {l1 l2 : Level} {A : UU l1} {B : UU l2}
+  where
+
+  is-not-injective : (A → B) → UU (l1 ⊔ l2)
+  is-not-injective f = ¬ (is-injective f)
+
+  is-prop-is-not-injective : {f : A → B} → is-prop (is-not-injective f)
+  is-prop-is-not-injective = is-prop-neg
+
+  is-not-injective-Prop : (A → B) → Prop (l1 ⊔ l2)
+  is-not-injective-Prop f = (is-not-injective f , is-prop-is-not-injective)
+```
+
 ## Properties
 
 ### Noninjective maps are not injective

From e03d7f73150271306b656b0d5cb33ea3ad5a8862 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Fri, 19 Sep 2025 07:48:49 +0200
Subject: [PATCH 35/41] fix `WDID` in `left-modules-rings`

---
 src/linear-algebra/left-modules-rings.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/linear-algebra/left-modules-rings.lagda.md b/src/linear-algebra/left-modules-rings.lagda.md
index 35ae1ca391..5b1b5daf21 100644
--- a/src/linear-algebra/left-modules-rings.lagda.md
+++ b/src/linear-algebra/left-modules-rings.lagda.md
@@ -36,7 +36,7 @@ open import ring-theory.rings
 ## Idea
 
 A
-{{#concept "left module" WD="left module" WDID="Q120721996" Agda=left-module-Ring}}
+{{#concept "left module" WD="left module" WDID=Q120721996 Agda=left-module-Ring}}
 `M` over a [ring](ring-theory.rings.md) `R` consists of an
 [abelian group](group-theory.abelian-groups.md) `M` equipped with an action
 `R → M → M` such that

From 055567e000b40ead863cad43a03d674fa3b6319e Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Fri, 19 Sep 2025 17:59:34 +0200
Subject: [PATCH 36/41] edits

---
 ...ormations-polynomial-endofunctors.lagda.md |  6 +++---
 ...morphisms-polynomial-endofunctors.lagda.md | 20 +++++++++++++++----
 ...univalent-polynomial-endofunctors.lagda.md |  5 +++--
 3 files changed, 22 insertions(+), 9 deletions(-)

diff --git a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
index 3d6f1b34c8..d5f00d7ba0 100644
--- a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
@@ -11,23 +11,22 @@ open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
 open import foundation.binary-homotopies
 open import foundation.cartesian-morphisms-arrows
-open import foundation.homotopies-morphisms-arrows
 open import foundation.commuting-squares-of-homotopies
 open import foundation.commuting-squares-of-maps
 open import foundation.cones-over-cospan-diagrams
 open import foundation.contractible-types
-open import foundation.lifts-morphisms-arrows
 open import foundation.dependent-pair-types
 open import foundation.equality-dependent-function-types
-open import foundation.universal-property-cartesian-morphisms-arrows
 open import foundation.equivalences
 open import foundation.function-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
+open import foundation.homotopies-morphisms-arrows
 open import foundation.homotopy-induction
 open import foundation.identity-types
 open import foundation.implicit-function-types
+open import foundation.lifts-morphisms-arrows
 open import foundation.morphisms-arrows
 open import foundation.precomposition-functions
 open import foundation.propositions
@@ -37,6 +36,7 @@ open import foundation.sections
 open import foundation.structure-identity-principle
 open import foundation.transport-along-identifications
 open import foundation.unit-type
+open import foundation.universal-property-cartesian-morphisms-arrows
 open import foundation.universe-levels
 open import foundation.whiskering-homotopies-composition
 
diff --git a/src/trees/morphisms-polynomial-endofunctors.lagda.md b/src/trees/morphisms-polynomial-endofunctors.lagda.md
index 98242f0b78..e9245a5b00 100644
--- a/src/trees/morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/morphisms-polynomial-endofunctors.lagda.md
@@ -27,6 +27,7 @@ open import foundation.identity-types
 open import foundation.implicit-function-types
 open import foundation.morphisms-arrows
 open import foundation.precomposition-functions
+open import foundation.retracts-of-types
 open import foundation.sections
 open import foundation.structure-identity-principle
 open import foundation.transport-along-identifications
@@ -355,6 +356,9 @@ module _
 
 ### Comparison between morphisms and natural transformations
 
+Morphisms of polynomial endofunctors form a retract of natural transformations,
+and this map is a section on shapes.
+
 ```agda
 module _
   {l1 l2 l3 l4 : Level}
@@ -380,14 +384,22 @@ module _
     inv
       ( naturality-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α x
         ( a , id))
+
+  retract-hom-natural-transformation-polynomial-endofunctor :
+    ( hom-polynomial-endofunctor 𝑃 𝑄) retract-of
+    ( natural-transformation-polynomial-endofunctor l2 𝑃 𝑄)
+  retract-hom-natural-transformation-polynomial-endofunctor =
+    ( λ f → natural-transformation-hom-polynomial-endofunctor 𝑃 𝑄 f {l2}) ,
+    ( hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄) ,
+    ( is-retraction-hom-natural-transformation-polynomial-endofunctor)
 ```
 
-**Comment.** If we were to show these notions are equivalent, we would have
-needed natural transformations to satisfy the following equality:
+**Comment.** If these notions were to be equivalent we would have needed natural
+transformations to satisfy the following equality:
 
 $$
   α₁ (f ∘ x) (a , id) = ap (𝑄 f) (α₁ x (a , id)) ∙ α₁ f (a , x),
 $$
 
-which is an unfolding of the condition that the naturality square of a composite
-map is given by pasting of squares.
+which is an instance of the unfolded condition that the naturality square of a
+composite map is given by pasting of squares.
diff --git a/src/trees/univalent-polynomial-endofunctors.lagda.md b/src/trees/univalent-polynomial-endofunctors.lagda.md
index 62a9e61f01..85e0a9b494 100644
--- a/src/trees/univalent-polynomial-endofunctors.lagda.md
+++ b/src/trees/univalent-polynomial-endofunctors.lagda.md
@@ -246,8 +246,9 @@ module _
 
   coh-refl-eq-Eq-type-univalent-polynomial-endofunctor :
     (x : type-univalent-polynomial-endofunctor 𝑃 X) →
-    ( eq-Eq-type-univalent-polynomial-endofunctor x x
-      ( refl-Eq-type-univalent-polynomial-endofunctor x)) ＝ refl
+    eq-Eq-type-univalent-polynomial-endofunctor x x
+      ( refl-Eq-type-univalent-polynomial-endofunctor x) ＝
+    refl
   coh-refl-eq-Eq-type-univalent-polynomial-endofunctor x =
     is-retraction-eq-Eq-type-univalent-polynomial-endofunctor x x refl
 ```

From 537a135d414be0c46877d95f3b0e5898604431b3 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 20 Sep 2025 15:19:22 +0200
Subject: [PATCH 37/41] A natural transformation into a polynomial endofunctor
 with a set of shapes is cartesian if and only if it is cartesian at terminal
 maps

---
 ...ormations-polynomial-endofunctors.lagda.md | 44 +++++++++++++++++--
 1 file changed, 41 insertions(+), 3 deletions(-)

diff --git a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
index d5f00d7ba0..36353ab2da 100644
--- a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
@@ -14,6 +14,7 @@ open import foundation.cartesian-morphisms-arrows
 open import foundation.commuting-squares-of-homotopies
 open import foundation.commuting-squares-of-maps
 open import foundation.cones-over-cospan-diagrams
+open import foundation.sets
 open import foundation.contractible-types
 open import foundation.dependent-pair-types
 open import foundation.equality-dependent-function-types
@@ -281,11 +282,12 @@ module _
       is-cartesian-comp-cartesian-natural-transformation-polynomial-endofunctor)
 ```
 
-### A natural transformation is cartesian if and only if it is cartesian at terminal maps
+### A natural transformation into a polynomial endofunctor with a set of shapes is cartesian if and only if it is cartesian at terminal maps
 
 **Proof.** One direction is trivial. For the other direction, given a natural
 transformation of polynomial endofunctors $α : 𝑃 ⇒ 𝑄$ and an arbitrary function
-$f : X → Y$, we have a morphism of arrows in the slice above $α_{*}$:
+$f : X → Y$, since the type of shapes of $𝑄$ is a set, the following prism
+commutes and we have a morphism of arrows in the slice above $α_{*}$:
 
 ```text
          αX
@@ -301,7 +303,43 @@ $f : X → Y$, we have a morphism of arrows in the slice above $α_{*}$:
 and so by the right-cancellation property of cartesian squares the naturality
 square at $f$ is cartesian. ∎
 
-This is mentioned as Remark 2.1.4 in {{#cite GHK22}}.
+This holds more generally for coherent natural transformations between arbitrary
+polynomial functors, as mentioned in Remark 2.1.4 in {{#cite GHK22}}.
+
+```agda
+module _
+  {l1 l2 l3 l4 l : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : natural-transformation-polynomial-endofunctor l 𝑃 𝑄)
+  where
+
+  is-cartesian-is-cartesian-at-terminal-map-natural-transformation-is-set-shape-polynomial-endofunctor :
+    is-set (shape-polynomial-endofunctor 𝑄) →
+    is-cartesian-at-terminal-map-natural-transformation-polynomial-endofunctor
+      𝑃 𝑄 α →
+    is-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α
+  is-cartesian-is-cartesian-at-terminal-map-natural-transformation-is-set-shape-polynomial-endofunctor
+    H𝑄 Hα {X} {Y} f =
+    is-pullback-top-square-vertical-triangle
+      (type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α)
+      (map-polynomial-endofunctor 𝑄 (raise-terminal-map Y))
+      (map-polynomial-endofunctor 𝑄 f)
+      (map-polynomial-endofunctor 𝑄 (raise-terminal-map X))
+      ( cone-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α
+        ( raise-terminal-map Y))
+      ( cone-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α f)
+      ( cone-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α
+        ( raise-terminal-map X))
+      ( refl-htpy)
+      ( refl-htpy ,
+        refl-htpy ,
+        λ x →
+        eq-is-prop
+          ( is-set-Σ H𝑄 (λ _ → is-set-function-type is-set-raise-unit) _ _))
+      ( Hα {Y})
+      ( Hα {X})
+```
 
 ## References
 

From 55e796e6aad02a9d18a550270c804b69573416cb Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 20 Sep 2025 15:36:33 +0200
Subject: [PATCH 38/41] fix another naming

---
 ...morphisms-polynomial-endofunctors.lagda.md | 44 +++++++++----------
 ...morphisms-polynomial-endofunctors.lagda.md | 44 +++++++++----------
 ...univalent-polynomial-endofunctors.lagda.md | 26 +++++------
 3 files changed, 57 insertions(+), 57 deletions(-)

diff --git a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
index 6e7652ad50..7899057662 100644
--- a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
@@ -83,14 +83,14 @@ module _
   is-cartesian-hom-polynomial-endofunctor : UU (l1 ⊔ l2 ⊔ l4)
   is-cartesian-hom-polynomial-endofunctor =
     (a : shape-polynomial-endofunctor 𝑃) →
-    is-equiv (positions-hom-polynomial-endofunctor 𝑃 𝑄 α a)
+    is-equiv (position-hom-polynomial-endofunctor 𝑃 𝑄 α a)
 
   is-prop-is-cartesian-hom-polynomial-endofunctor :
     is-prop is-cartesian-hom-polynomial-endofunctor
   is-prop-is-cartesian-hom-polynomial-endofunctor =
     is-prop-Π
       ( λ a →
-        is-property-is-equiv (positions-hom-polynomial-endofunctor 𝑃 𝑄 α a))
+        is-property-is-equiv (position-hom-polynomial-endofunctor 𝑃 𝑄 α a))
 
   is-cartesian-hom-polynomial-endofunctor-Prop : Prop (l1 ⊔ l2 ⊔ l4)
   is-cartesian-hom-polynomial-endofunctor-Prop =
@@ -132,21 +132,21 @@ module _
   hom-cartesian-hom-polynomial-endofunctor : hom-polynomial-endofunctor 𝑃 𝑄
   hom-cartesian-hom-polynomial-endofunctor = pr1 α
 
-  shapes-cartesian-hom-polynomial-endofunctor :
+  shape-cartesian-hom-polynomial-endofunctor :
     shape-polynomial-endofunctor 𝑃 → shape-polynomial-endofunctor 𝑄
-  shapes-cartesian-hom-polynomial-endofunctor =
-    shapes-hom-polynomial-endofunctor 𝑃 𝑄
+  shape-cartesian-hom-polynomial-endofunctor =
+    shape-hom-polynomial-endofunctor 𝑃 𝑄
       hom-cartesian-hom-polynomial-endofunctor
 
-  positions-cartesian-hom-polynomial-endofunctor :
+  position-cartesian-hom-polynomial-endofunctor :
     (a : shape-polynomial-endofunctor 𝑃) →
     position-polynomial-endofunctor 𝑄
-      ( shapes-hom-polynomial-endofunctor 𝑃 𝑄
+      ( shape-hom-polynomial-endofunctor 𝑃 𝑄
         ( hom-cartesian-hom-polynomial-endofunctor)
         ( a)) →
     position-polynomial-endofunctor 𝑃 a
-  positions-cartesian-hom-polynomial-endofunctor =
-    positions-hom-polynomial-endofunctor 𝑃 𝑄
+  position-cartesian-hom-polynomial-endofunctor =
+    position-hom-polynomial-endofunctor 𝑃 𝑄
       hom-cartesian-hom-polynomial-endofunctor
 
   type-cartesian-hom-polynomial-endofunctor :
@@ -161,15 +161,15 @@ module _
       hom-cartesian-hom-polynomial-endofunctor
   is-cartesian-cartesian-hom-polynomial-endofunctor = pr2 α
 
-  equiv-positions-cartesian-hom-polynomial-endofunctor :
+  equiv-position-cartesian-hom-polynomial-endofunctor :
     (a : shape-polynomial-endofunctor 𝑃) →
     position-polynomial-endofunctor 𝑄
-      ( shapes-hom-polynomial-endofunctor 𝑃 𝑄
+      ( shape-hom-polynomial-endofunctor 𝑃 𝑄
         ( hom-cartesian-hom-polynomial-endofunctor)
         ( a)) ≃
     position-polynomial-endofunctor 𝑃 a
-  equiv-positions-cartesian-hom-polynomial-endofunctor a =
-    ( positions-cartesian-hom-polynomial-endofunctor a ,
+  equiv-position-cartesian-hom-polynomial-endofunctor a =
+    ( position-cartesian-hom-polynomial-endofunctor a ,
       is-cartesian-cartesian-hom-polynomial-endofunctor a)
 
   naturality-cartesian-hom-polynomial-endofunctor :
@@ -308,8 +308,8 @@ module _
   (𝑃 : polynomial-endofunctor l1 l2)
   (𝑄 : polynomial-endofunctor l3 l4)
   (α : cartesian-hom-polynomial-endofunctor 𝑃 𝑄)
-  (let α₀ = shapes-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α)
-  (let α₁ = positions-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α)
+  (let α₀ = shape-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α)
+  (let α₁ = position-cartesian-hom-polynomial-endofunctor 𝑃 𝑄 α)
   {X : UU l5}
   where
 
@@ -393,20 +393,20 @@ module _
   (let α₀ = type-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α)
   where
 
-  shapes-cartesian-natural-transformation-polynomial-endofunctor :
+  shape-cartesian-natural-transformation-polynomial-endofunctor :
     shape-polynomial-endofunctor 𝑃 → shape-polynomial-endofunctor 𝑄
-  shapes-cartesian-natural-transformation-polynomial-endofunctor =
-    shapes-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+  shape-cartesian-natural-transformation-polynomial-endofunctor =
+    shape-natural-transformation-polynomial-endofunctor 𝑃 𝑄
       ( natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
           𝑃 𝑄 α)
 
-  positions-cartesian-natural-transformation-polynomial-endofunctor :
+  position-cartesian-natural-transformation-polynomial-endofunctor :
     (a : shape-polynomial-endofunctor 𝑃) →
     position-polynomial-endofunctor 𝑄
-      ( shapes-cartesian-natural-transformation-polynomial-endofunctor a) →
+      ( shape-cartesian-natural-transformation-polynomial-endofunctor a) →
     position-polynomial-endofunctor 𝑃 a
-  positions-cartesian-natural-transformation-polynomial-endofunctor =
-    positions-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+  position-cartesian-natural-transformation-polynomial-endofunctor =
+    position-natural-transformation-polynomial-endofunctor 𝑃 𝑄
       ( natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
           𝑃 𝑄 α)
 
diff --git a/src/trees/morphisms-polynomial-endofunctors.lagda.md b/src/trees/morphisms-polynomial-endofunctors.lagda.md
index e9245a5b00..a02d3c8b15 100644
--- a/src/trees/morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/morphisms-polynomial-endofunctors.lagda.md
@@ -73,15 +73,15 @@ module _
   (α : hom-polynomial-endofunctor 𝑃 𝑄)
   where
 
-  shapes-hom-polynomial-endofunctor :
+  shape-hom-polynomial-endofunctor :
     shape-polynomial-endofunctor 𝑃 → shape-polynomial-endofunctor 𝑄
-  shapes-hom-polynomial-endofunctor = pr1 α
+  shape-hom-polynomial-endofunctor = pr1 α
 
-  positions-hom-polynomial-endofunctor :
+  position-hom-polynomial-endofunctor :
     (a : shape-polynomial-endofunctor 𝑃) →
-    position-polynomial-endofunctor 𝑄 (shapes-hom-polynomial-endofunctor a) →
+    position-polynomial-endofunctor 𝑄 (shape-hom-polynomial-endofunctor a) →
     position-polynomial-endofunctor 𝑃 a
-  positions-hom-polynomial-endofunctor = pr2 α
+  position-hom-polynomial-endofunctor = pr2 α
 
   type-hom-polynomial-endofunctor :
     {l3 : Level} {X : UU l3} →
@@ -90,8 +90,8 @@ module _
   type-hom-polynomial-endofunctor {X = X} =
     map-Σ
       ( λ c → position-polynomial-endofunctor 𝑄 c → X)
-      ( shapes-hom-polynomial-endofunctor)
-      ( λ a → precomp (positions-hom-polynomial-endofunctor a) X)
+      ( shape-hom-polynomial-endofunctor)
+      ( λ a → precomp (position-hom-polynomial-endofunctor a) X)
 ```
 
 ### The identity morphism
@@ -133,15 +133,15 @@ module _
   htpy-hom-polynomial-endofunctor :
     (α β : hom-polynomial-endofunctor 𝑃 𝑄) → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
   htpy-hom-polynomial-endofunctor α β =
-    Σ ( shapes-hom-polynomial-endofunctor 𝑃 𝑄 α ~
-        shapes-hom-polynomial-endofunctor 𝑃 𝑄 β)
+    Σ ( shape-hom-polynomial-endofunctor 𝑃 𝑄 α ~
+        shape-hom-polynomial-endofunctor 𝑃 𝑄 β)
       ( λ H →
         (a : shape-polynomial-endofunctor 𝑃)
         (d :
           position-polynomial-endofunctor 𝑄
-            ( shapes-hom-polynomial-endofunctor 𝑃 𝑄 α a)) →
-        positions-hom-polynomial-endofunctor 𝑃 𝑄 α a d ＝
-        positions-hom-polynomial-endofunctor 𝑃 𝑄 β a
+            ( shape-hom-polynomial-endofunctor 𝑃 𝑄 α a)) →
+        position-hom-polynomial-endofunctor 𝑃 𝑄 α a d ＝
+        position-hom-polynomial-endofunctor 𝑃 𝑄 β a
           ( tr (position-polynomial-endofunctor 𝑄) (H a) d))
 
   refl-htpy-hom-polynomial-endofunctor :
@@ -159,9 +159,9 @@ module _
     is-torsorial (htpy-hom-polynomial-endofunctor α)
   is-torsorial-htpy-hom-polynomial-endofunctor α =
     is-torsorial-Eq-structure
-      ( is-torsorial-htpy (shapes-hom-polynomial-endofunctor 𝑃 𝑄 α))
-      ( shapes-hom-polynomial-endofunctor 𝑃 𝑄 α , refl-htpy)
-      ( is-torsorial-binary-htpy (positions-hom-polynomial-endofunctor 𝑃 𝑄 α))
+      ( is-torsorial-htpy (shape-hom-polynomial-endofunctor 𝑃 𝑄 α))
+      ( shape-hom-polynomial-endofunctor 𝑃 𝑄 α , refl-htpy)
+      ( is-torsorial-binary-htpy (position-hom-polynomial-endofunctor 𝑃 𝑄 α))
 
   is-equiv-htpy-eq-hom-polynomial-endofunctor :
     (α β : hom-polynomial-endofunctor 𝑃 𝑄) →
@@ -266,24 +266,24 @@ module _
   (α : natural-transformation-polynomial-endofunctor l2 𝑃 𝑄)
   where
 
-  shapes-natural-transformation-polynomial-endofunctor :
+  shape-natural-transformation-polynomial-endofunctor :
     shape-polynomial-endofunctor 𝑃 → shape-polynomial-endofunctor 𝑄
-  shapes-natural-transformation-polynomial-endofunctor a =
+  shape-natural-transformation-polynomial-endofunctor a =
     pr1 (type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α (a , id))
 
-  positions-natural-transformation-polynomial-endofunctor :
+  position-natural-transformation-polynomial-endofunctor :
     (a : shape-polynomial-endofunctor 𝑃) →
     position-polynomial-endofunctor 𝑄
-      ( shapes-natural-transformation-polynomial-endofunctor a) →
+      ( shape-natural-transformation-polynomial-endofunctor a) →
     position-polynomial-endofunctor 𝑃 a
-  positions-natural-transformation-polynomial-endofunctor a =
+  position-natural-transformation-polynomial-endofunctor a =
     pr2 (type-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α (a , id))
 
   hom-natural-transformation-polynomial-endofunctor :
     hom-polynomial-endofunctor 𝑃 𝑄
   hom-natural-transformation-polynomial-endofunctor =
-    ( shapes-natural-transformation-polynomial-endofunctor ,
-      positions-natural-transformation-polynomial-endofunctor)
+    ( shape-natural-transformation-polynomial-endofunctor ,
+      position-natural-transformation-polynomial-endofunctor)
 ```
 
 ### Computing the fibers of the induced natural transformation
diff --git a/src/trees/univalent-polynomial-endofunctors.lagda.md b/src/trees/univalent-polynomial-endofunctors.lagda.md
index 85e0a9b494..1a59902f1f 100644
--- a/src/trees/univalent-polynomial-endofunctors.lagda.md
+++ b/src/trees/univalent-polynomial-endofunctors.lagda.md
@@ -88,14 +88,14 @@ module _
     polynomial-endofunctor l1 l2
   polynomial-endofunctor-univalent-polynomial-endofunctor = pr1 𝑃
 
-  shapes-univalent-polynomial-endofunctor : UU l1
-  shapes-univalent-polynomial-endofunctor =
+  shape-univalent-polynomial-endofunctor : UU l1
+  shape-univalent-polynomial-endofunctor =
     shape-polynomial-endofunctor
       polynomial-endofunctor-univalent-polynomial-endofunctor
 
-  positions-univalent-polynomial-endofunctor :
-    shapes-univalent-polynomial-endofunctor → UU l2
-  positions-univalent-polynomial-endofunctor =
+  position-univalent-polynomial-endofunctor :
+    shape-univalent-polynomial-endofunctor → UU l2
+  position-univalent-polynomial-endofunctor =
     position-polynomial-endofunctor
       polynomial-endofunctor-univalent-polynomial-endofunctor
 
@@ -105,16 +105,16 @@ module _
   is-univalent-univalent-polynomial-endofunctor = pr2 𝑃
 
   univalent-family-univalent-polynomial-endofunctor :
-    univalent-family l2 shapes-univalent-polynomial-endofunctor
+    univalent-family l2 shape-univalent-polynomial-endofunctor
   univalent-family-univalent-polynomial-endofunctor =
-    ( positions-univalent-polynomial-endofunctor ,
+    ( position-univalent-polynomial-endofunctor ,
       is-univalent-univalent-polynomial-endofunctor)
 
   equiv-equiv-tr-univalent-polynomial-endofunctor :
-    {x y : shapes-univalent-polynomial-endofunctor} →
+    {x y : shape-univalent-polynomial-endofunctor} →
     ( x ＝ y) ≃
-    ( positions-univalent-polynomial-endofunctor x ≃
-      positions-univalent-polynomial-endofunctor y)
+    ( position-univalent-polynomial-endofunctor x ≃
+      position-univalent-polynomial-endofunctor y)
   equiv-equiv-tr-univalent-polynomial-endofunctor =
     equiv-equiv-tr-univalent-family
       univalent-family-univalent-polynomial-endofunctor
@@ -185,8 +185,8 @@ module _
   Eq-type-univalent-polynomial-endofunctor :
     (x y : type-univalent-polynomial-endofunctor 𝑃 X) → UU (l2 ⊔ l3)
   Eq-type-univalent-polynomial-endofunctor x y =
-    Σ ( positions-univalent-polynomial-endofunctor 𝑃 (pr1 x) ≃
-        positions-univalent-polynomial-endofunctor 𝑃 (pr1 y))
+    Σ ( position-univalent-polynomial-endofunctor 𝑃 (pr1 x) ≃
+        position-univalent-polynomial-endofunctor 𝑃 (pr1 y))
       ( λ e → coherence-triangle-maps (pr2 x) (pr2 y) (map-equiv e))
 
   refl-Eq-type-univalent-polynomial-endofunctor :
@@ -207,7 +207,7 @@ module _
     is-torsorial-Eq-type-univalent-polynomial-endofunctor (x , α) =
       is-torsorial-Eq-structure
         ( is-contr-equiv'
-          ( Σ (shapes-univalent-polynomial-endofunctor 𝑃) (x ＝_))
+          ( Σ (shape-univalent-polynomial-endofunctor 𝑃) (x ＝_))
           ( equiv-tot
             ( λ y → equiv-equiv-tr-univalent-polynomial-endofunctor 𝑃 {x} {y}))
           ( is-torsorial-Id x))

From ed04aa0fecbddc4fa7e05a8d8e98a21f5943da3d Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 20 Sep 2025 15:49:05 +0200
Subject: [PATCH 39/41] pre-commit

---
 .../universal-property-cartesian-morphisms-arrows.lagda.md    | 4 +++-
 ...n-natural-transformations-polynomial-endofunctors.lagda.md | 2 +-
 2 files changed, 4 insertions(+), 2 deletions(-)

diff --git a/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md b/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
index 3c9301ddd8..fa6b534186 100644
--- a/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
+++ b/src/foundation/universal-property-cartesian-morphisms-arrows.lagda.md
@@ -185,7 +185,9 @@ module _
         ( equiv-tot
           ( equiv-htpy-cone-is-lift-hom-arrow-of-lift-codomain-hom-arrow
             f g h α β (i , I)))
-        ( uniqueness-universal-property-pullback (map-codomain-hom-arrow g h β) h
+        ( uniqueness-universal-property-pullback
+          ( map-codomain-hom-arrow g h β)
+          ( h)
           ( cone-hom-arrow g h β)
           ( up-pullback-cartesian-hom-arrow g h (β , H))
           ( A)
diff --git a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
index 36353ab2da..626c2305fe 100644
--- a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
@@ -14,7 +14,6 @@ open import foundation.cartesian-morphisms-arrows
 open import foundation.commuting-squares-of-homotopies
 open import foundation.commuting-squares-of-maps
 open import foundation.cones-over-cospan-diagrams
-open import foundation.sets
 open import foundation.contractible-types
 open import foundation.dependent-pair-types
 open import foundation.equality-dependent-function-types
@@ -34,6 +33,7 @@ open import foundation.propositions
 open import foundation.pullbacks
 open import foundation.raising-universe-levels
 open import foundation.sections
+open import foundation.sets
 open import foundation.structure-identity-principle
 open import foundation.transport-along-identifications
 open import foundation.unit-type

From d0bfc8ce9c4d08333490b824cf610127f754c5f6 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 20 Sep 2025 16:45:23 +0200
Subject: [PATCH 40/41] edits

---
 .../univalent-type-families.lagda.md          |  2 +-
 ...morphisms-polynomial-endofunctors.lagda.md | 42 +------------------
 ...ormations-polynomial-endofunctors.lagda.md | 39 +++++++++++++++++
 ...morphisms-polynomial-endofunctors.lagda.md |  4 ++
 4 files changed, 46 insertions(+), 41 deletions(-)

diff --git a/src/foundation/univalent-type-families.lagda.md b/src/foundation/univalent-type-families.lagda.md
index 1d11324a68..3ffe7fdd2e 100644
--- a/src/foundation/univalent-type-families.lagda.md
+++ b/src/foundation/univalent-type-families.lagda.md
@@ -262,7 +262,7 @@ module _
     (type-family-univalent-family ℬ a , a , id-equiv)
 
   map-inv-equiv-type-subuniverse-univalent-family :
-     type-subuniverse (subuniverse-univalent-family ℬ l2) → A
+    type-subuniverse (subuniverse-univalent-family ℬ l2) → A
   map-inv-equiv-type-subuniverse-univalent-family (X , a , e) = a
 
   is-retraction-map-inv-equiv-type-subuniverse-univalent-family :
diff --git a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
index 7899057662..5777560a84 100644
--- a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
@@ -11,6 +11,8 @@ open import foundation.action-on-identifications-functions
 open import foundation.binary-homotopies
 open import foundation.commuting-squares-of-homotopies
 open import foundation.commuting-squares-of-maps
+open import foundation.raising-universe-levels
+open import foundation.unit-type
 open import foundation.cones-over-cospan-diagrams
 open import foundation.contractible-maps
 open import foundation.contractible-types
@@ -380,46 +382,6 @@ module _
             ( is-equiv-id))
 ```
 
-### The associated morphism of a cartesian natural transformation is cartesian
-
-```agda
-module _
-  {l1 l2 l3 l4 : Level}
-  (𝑃 : polynomial-endofunctor l1 l2)
-  (𝑄 : polynomial-endofunctor l3 l4)
-  (α : cartesian-natural-transformation-polynomial-endofunctor l2 𝑃 𝑄)
-  (let 𝑃₀ = shape-polynomial-endofunctor 𝑃)
-  (let 𝑃₁ = position-polynomial-endofunctor 𝑃)
-  (let α₀ = type-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α)
-  where
-
-  shape-cartesian-natural-transformation-polynomial-endofunctor :
-    shape-polynomial-endofunctor 𝑃 → shape-polynomial-endofunctor 𝑄
-  shape-cartesian-natural-transformation-polynomial-endofunctor =
-    shape-natural-transformation-polynomial-endofunctor 𝑃 𝑄
-      ( natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
-          𝑃 𝑄 α)
-
-  position-cartesian-natural-transformation-polynomial-endofunctor :
-    (a : shape-polynomial-endofunctor 𝑃) →
-    position-polynomial-endofunctor 𝑄
-      ( shape-cartesian-natural-transformation-polynomial-endofunctor a) →
-    position-polynomial-endofunctor 𝑃 a
-  position-cartesian-natural-transformation-polynomial-endofunctor =
-    position-natural-transformation-polynomial-endofunctor 𝑃 𝑄
-      ( natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
-          𝑃 𝑄 α)
-
-  hom-cartesian-natural-transformation-polynomial-endofunctor :
-    hom-polynomial-endofunctor 𝑃 𝑄
-  hom-cartesian-natural-transformation-polynomial-endofunctor =
-    hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄
-      ( natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
-          𝑃 𝑄 α)
-```
-
-> TODO
-
 ### Equivalence between cartesian morphisms and cartesian natural transformations
 
 This is proved in a slightly different setting as part of Lemma 2.3.3 in
diff --git a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
index 626c2305fe..988cdf7ae3 100644
--- a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
@@ -47,6 +47,7 @@ open import foundation-core.torsorial-type-families
 
 open import trees.natural-transformations-polynomial-endofunctors
 open import trees.polynomial-endofunctors
+open import trees.morphisms-polynomial-endofunctors
 ```
 
 </details>
@@ -238,6 +239,44 @@ module _
     ( hom-arrow-cartesian-natural-transformation-polynomial-endofunctor f , H f)
 ```
 
+### The associated morphism of polynomial endofunctors
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (𝑃 : polynomial-endofunctor l1 l2)
+  (𝑄 : polynomial-endofunctor l3 l4)
+  (α : cartesian-natural-transformation-polynomial-endofunctor l2 𝑃 𝑄)
+  (let 𝑃₀ = shape-polynomial-endofunctor 𝑃)
+  (let 𝑃₁ = position-polynomial-endofunctor 𝑃)
+  (let α₀ = type-cartesian-natural-transformation-polynomial-endofunctor 𝑃 𝑄 α)
+  where
+
+  shape-cartesian-natural-transformation-polynomial-endofunctor :
+    shape-polynomial-endofunctor 𝑃 → shape-polynomial-endofunctor 𝑄
+  shape-cartesian-natural-transformation-polynomial-endofunctor =
+    shape-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
+          𝑃 𝑄 α)
+
+  position-cartesian-natural-transformation-polynomial-endofunctor :
+    (a : shape-polynomial-endofunctor 𝑃) →
+    position-polynomial-endofunctor 𝑄
+      ( shape-cartesian-natural-transformation-polynomial-endofunctor a) →
+    position-polynomial-endofunctor 𝑃 a
+  position-cartesian-natural-transformation-polynomial-endofunctor =
+    position-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
+          𝑃 𝑄 α)
+
+  hom-cartesian-natural-transformation-polynomial-endofunctor :
+    hom-polynomial-endofunctor 𝑃 𝑄
+  hom-cartesian-natural-transformation-polynomial-endofunctor =
+    hom-natural-transformation-polynomial-endofunctor 𝑃 𝑄
+      ( natural-transformation-cartesian-natural-transformation-polynomial-endofunctor
+          𝑃 𝑄 α)
+```
+
 ### The identity cartesian natural transformation
 
 ```agda
diff --git a/src/trees/morphisms-polynomial-endofunctors.lagda.md b/src/trees/morphisms-polynomial-endofunctors.lagda.md
index a02d3c8b15..9e3ff069f3 100644
--- a/src/trees/morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/morphisms-polynomial-endofunctors.lagda.md
@@ -258,6 +258,10 @@ module _
 
 ### Natural transformations define morphisms
 
+Given a natural transformation `α : 𝑃 ⇒ 𝑄` then we have an associated morphism
+given on shapes by `a ↦ pr1 (α₀ {𝑃₁ a} (a , id)) : 𝑃₀ → 𝑄₀` and on positions by
+`a ↦ pr2 (α₀ {𝑃₁ a} (a , id)) : (a : 𝑃₀) → 𝑄₁ _ → 𝑃₁ a`.
+
 ```agda
 module _
   {l1 l2 l3 l4 : Level}

From 101235666c495747666f92b304a202cd0d8a6af3 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sat, 20 Sep 2025 16:45:32 +0200
Subject: [PATCH 41/41] edits

---
 .../cartesian-morphisms-polynomial-endofunctors.lagda.md      | 4 ++--
 ...n-natural-transformations-polynomial-endofunctors.lagda.md | 2 +-
 2 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
index 5777560a84..b6f1ed0338 100644
--- a/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-morphisms-polynomial-endofunctors.lagda.md
@@ -11,8 +11,6 @@ open import foundation.action-on-identifications-functions
 open import foundation.binary-homotopies
 open import foundation.commuting-squares-of-homotopies
 open import foundation.commuting-squares-of-maps
-open import foundation.raising-universe-levels
-open import foundation.unit-type
 open import foundation.cones-over-cospan-diagrams
 open import foundation.contractible-maps
 open import foundation.contractible-types
@@ -34,6 +32,7 @@ open import foundation.morphisms-arrows
 open import foundation.precomposition-functions
 open import foundation.propositions
 open import foundation.pullbacks
+open import foundation.raising-universe-levels
 open import foundation.sections
 open import foundation.structure-identity-principle
 open import foundation.subtypes
@@ -42,6 +41,7 @@ open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.type-theoretic-principle-of-choice
+open import foundation.unit-type
 open import foundation.universal-property-equivalences
 open import foundation.universe-levels
 open import foundation.whiskering-homotopies-composition
diff --git a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
index 988cdf7ae3..f679d5d5f7 100644
--- a/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
+++ b/src/trees/cartesian-natural-transformations-polynomial-endofunctors.lagda.md
@@ -45,9 +45,9 @@ open import foundation-core.equality-dependent-pair-types
 open import foundation-core.retractions
 open import foundation-core.torsorial-type-families
 
+open import trees.morphisms-polynomial-endofunctors
 open import trees.natural-transformations-polynomial-endofunctors
 open import trees.polynomial-endofunctors
-open import trees.morphisms-polynomial-endofunctors
 ```
 
 </details>