Definition preserves_comp_cat_dependent_prod_iso {C₁ C₂ : full_comp_cat} (F : full_comp_cat_functor C₁ C₂) (P₁ : comp_cat_dependent_prod C₁) (P₂ : comp_cat_dependent_prod C₂) (HF : preserves_comp_cat_dependent_prod F P₁ P₂) {Γ Δ : C₁} (A : ty Γ) (s : z_iso Δ (Γ & A)) (s' : Δ --> Γ) (p : s · π A = s') (B : ty Δ) (q : z_iso_comp (functor_on_z_iso F s) (full_comp_cat_functor_z_iso F A) · π (comp_cat_type_functor F Γ A) = # F s') : is_z_isomorphism (right_beck_chevalley_nat_trans (dep_prod_comp_cat_iso P₁ _ _ _ p) (dep_prod_comp_cat_iso P₂ _ _ _ q) (fiber_functor_natural_nat_z_iso (cleaving_of_types C₁) (cleaving_of_types C₂) (cartesian_comp_cat_type_functor F) _) B).
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.