Proposition comprehension_nat_trans_comm {C₁ C₂ : cat_with_terminal_cleaving} {χ₁ : comprehension_functor C₁} {χ₂ : comprehension_functor C₂} {F : functor_with_terminal_cleaving C₁ C₂} (Fχ : comprehension_nat_trans χ₁ χ₂ F) {x y : C₁} {f : x --> y} {xx : disp_cat_of_types C₁ x} {yy : disp_cat_of_types C₁ y} (ff : xx -->[ f ] yy) : #F (comprehension_functor_mor χ₁ ff) · comprehension_nat_trans_mor Fχ yy = comprehension_nat_trans_mor Fχ xx · comprehension_functor_mor χ₂ (♯ (comp_cat_type_functor F) ff). Proof. refine (maponpaths pr1 (pr2 Fχ x y f xx yy ff) @ _). unfold transportb. refine (pr1_transportf (A := C₂⟦_,_⟧) _ _ @ _). rewrite transportf_const ; cbn. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodFunctor. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq. Require Import UniMath.CategoryTheory.DisplayedCats.ComprehensionC. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.DispCatTerminal. Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.FibTerminal. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat.