Proposition cat_property_fiber_functor_comp (P : cat_property) {C₁ C₂ C₃ : dfl_full_comp_cat} {F : dfl_full_comp_cat_functor C₁ C₂} {G : dfl_full_comp_cat_functor C₂ C₃} {Γ : C₁} {p₁ : P (dfl_full_comp_cat_fiber Γ)} {p₂ : P (dfl_full_comp_cat_fiber (F Γ))} {p₃ : P (dfl_full_comp_cat_fiber (G(F Γ)))} (HF : cat_property_functor P p₁ p₂ (dfl_full_comp_cat_functor_fiber_functor F Γ)) (HG : cat_property_functor P p₂ p₃ (dfl_full_comp_cat_functor_fiber_functor G (F Γ))) : cat_property_functor P p₁ p₃ (dfl_full_comp_cat_functor_fiber_functor (F · G) Γ). Proof. use (cat_property_functor_nat_z_iso_f P _ (cat_property_comp_functor P HF HG)). apply fiber_functor_comp_nat_z_iso. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.Limits.PreservationProperties. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodFunctor. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodLimits. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Examples.StructuredCategories. Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat. Require Import UniMath.Bicategories.ComprehensionCat.DFLCompCat.