Definition fiber_functor_identity {C : category} (D : disp_cat C) (x : C) : nat_z_iso (functor_identity _) (fiber_functor (disp_functor_identity D) x). Proof. use make_nat_z_iso. - use make_nat_trans. + exact (λ _, identity _). + intros xx yy ff ; cbn. rewrite id_right_disp. rewrite id_left_disp. unfold transportb. rewrite !transport_f_f. apply maponpaths_2. apply homset_property. - intros xx. apply (@is_z_isomorphism_identity (fiber_category D x)). 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.