Definition indexed_cat_to_disp_cat_z_iso_weq_base {x : C} (xx₁ xx₂ : Φ x) : (xx₁ --> xx₂) ≃ (xx₁ --> (Φ $ identity x) xx₂). Proof. use weq_iso. - exact (λ f, f · indexed_cat_id Φ x xx₂). - exact (λ f, f · inv_from_z_iso (indexed_cat_id_z_iso Φ xx₂)). - abstract (intro f ; cbn ; rewrite !assoc' ; refine (_ @ id_right _) ; apply maponpaths ; exact (z_iso_inv_after_z_iso (indexed_cat_id_z_iso Φ xx₂))). - abstract (intro f ; cbn ; rewrite !assoc' ; refine (_ @ id_right _) ; apply maponpaths ; exact (z_iso_after_z_iso_inv (indexed_cat_id_z_iso Φ xx₂))). 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.