proposition: 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: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Codomain UniMath.CategoryTheory.DisplayedCats.Codomain.CodFunctor UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq UniMath.CategoryTheory.DisplayedCats.ComprehensionC UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.Preservation UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.DispCatTerminal UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.FibTerminal UniMath.Bicategories.DisplayedBicats.DispBicat UniMath.Bicategories.DisplayedBicats.DispAdjunctions UniMath.Bicategories.DisplayedBicats.DispInvertibles UniMath.Bicategories.DisplayedBicats.DispUnivalence UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat