Definition preserves_comp_cat_dependent_prod {C₁ C₂ : full_comp_cat} (F : full_comp_cat_functor C₁ C₂) (P₁ : comp_cat_dependent_prod C₁) (P₂ : comp_cat_dependent_prod C₂) : UU. Proof. simple refine (∏ (Γ : C₁) (A : ty Γ) (B : ty (Γ & A)), is_z_isomorphism (right_beck_chevalley_nat_trans (pr1 P₁ Γ A) _ (fiber_functor_natural_nat_z_iso (cleaving_of_types C₁) (cleaving_of_types C₂) (cartesian_comp_cat_type_functor F) _) B)). use (dep_prod_comp_cat_iso P₂). - exact (comp_cat_type_functor F Γ A). - exact (full_comp_cat_functor_z_iso F A). - apply (comprehension_nat_trans_mor_comm (comp_cat_functor_comprehension F)). Defined.
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.