Proposition indexed_cat_to_disp_cat_from_disp_iso_inv_right : f · indexed_cat_to_disp_cat_from_disp_iso_inv = identity _. Proof. unfold indexed_cat_to_disp_cat_from_disp_iso_inv. rewrite !assoc. refine (!_). use z_iso_inv_on_left. rewrite id_left. pose (inv_mor_after_z_iso_disp Hf) as p. cbn in p. unfold transportb in p. rewrite transportf_indexed_cat_to_disp_cat in p. use (cancel_z_iso _ _ (idtoiso (maponpaths (λ h, (Φ $ h) xx₁) (!(id_left (identity x)))))). refine (_ @ p) ; clear p. rewrite !assoc'. apply maponpaths. rewrite !assoc. refine (!_). etrans. { apply maponpaths_2. refine (!_). apply (nat_trans_ax (indexed_cat_id Φ x)). } cbn. rewrite !assoc'. apply maponpaths. apply (indexed_cat_runitor_alt Φ). Qed.
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.