Definition is_z_iso_disp_indexed_cat_to_disp_cat {x : C} (xx₁ xx₂ : Φ x) (f : xx₁ --> (Φ $ identity x) xx₂) (Hf : is_z_isomorphism f) : @is_z_iso_disp _ indexed_cat_to_disp_cat _ _ (identity_z_iso x) xx₁ xx₂ f. Proof. pose (f' := f · inv_from_z_iso (indexed_cat_id_z_iso Φ xx₂)). refine (transportf (λ z, is_z_iso_disp _ z) _ (indexed_cat_to_disp_cat_to_disp_iso xx₁ xx₂ f' _)). - cbn ; unfold f'. rewrite !assoc'. refine (_ @ id_right _). apply maponpaths. apply z_iso_after_z_iso_inv. - use is_z_isomorphism_comp. + exact Hf. + apply is_z_iso_inv_from_z_iso. Defined.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.OppositeCategory.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedCategory.