Proposition right_beck_chevalley_nat_trans_mor_eq {C₁ C₂ : full_comp_cat} (F : full_comp_cat_functor C₁ C₂) {Γ₁ Γ₂ : C₁} {s s' : Γ₁ --> Γ₂} (p : s = s') (P₁ : dependent_product (cleaving_of_types C₁) s) (P₂ : dependent_product (cleaving_of_types C₂) (#F s)) (P₁' : dependent_product (cleaving_of_types C₁) s') (P₂' : dependent_product (cleaving_of_types C₂) (#F s')) (B : ty Γ₁) (H : is_z_isomorphism (right_beck_chevalley_nat_trans P₁ P₂ (fiber_functor_natural_nat_z_iso (cleaving_of_types C₁) (cleaving_of_types C₂) (cartesian_comp_cat_type_functor F) _) B)) : is_z_isomorphism (right_beck_chevalley_nat_trans P₁' P₂' (fiber_functor_natural_nat_z_iso (cleaving_of_types C₁) (cleaving_of_types C₂) (cartesian_comp_cat_type_functor F) _) B). Proof. induction p. assert (P₁ = P₁') as ->. { apply isaprop_dependent_product. } assert (P₂ = P₂') as ->. { apply isaprop_dependent_product. } exact H. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.DependentProducts. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.UnivalenceOp. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Examples.OpMorBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.LiftDispBicat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell. Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat. Require Import UniMath.Bicategories.ComprehensionCat.DFLCompCat. Require Import UniMath.Bicategories.ComprehensionCat.CompCatNotations.