Definition fiber_functor_comp_nat_z_iso {C₁ C₂ C₃ : category} {F : C₁ ⟶ C₂} {G : C₂ ⟶ C₃} {D₁ : disp_cat C₁} {D₂ : disp_cat C₂} {D₃ : disp_cat C₃} (FF : disp_functor F D₁ D₂) (GG : disp_functor G D₂ D₃) (x : C₁) : nat_z_iso (fiber_functor FF x ∙ fiber_functor GG (F x)) (fiber_functor (disp_functor_composite FF GG) x). Proof. use make_nat_z_iso. - exact (fiber_functor_comp_nat_trans FF GG x). - intro xx. apply is_z_isomorphism_identity. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Total.