Proposition indexed_cat_to_disp_cat_from_disp_iso_inv_left : indexed_cat_to_disp_cat_from_disp_iso_inv · f = identity _. Proof. unfold indexed_cat_to_disp_cat_from_disp_iso_inv. rewrite !assoc'. etrans. { apply maponpaths. refine (!_). apply (nat_trans_ax (nat_z_iso_inv (indexed_cat_id_nat_z_iso Φ x)) _ _ _ : _ · inv_from_z_iso (indexed_cat_id_z_iso Φ xx₂) = inv_from_z_iso (indexed_cat_id_z_iso Φ xx₁) · f). } rewrite !assoc. refine (!_). use z_iso_inv_on_left. rewrite id_left. use (cancel_z_iso _ _ (idtoiso (maponpaths (λ h, (Φ $ h) xx₂) (!(id_left (identity x)))))). pose (z_iso_disp_after_inv_mor Hf) as p. cbn in p. unfold transportb in p. rewrite transportf_indexed_cat_to_disp_cat in p. refine (_ @ p) ; clear p. rewrite !functor_comp. rewrite !assoc'. do 2 apply maponpaths. refine (!_). refine (indexed_cat_lunitor_alt Φ _ _ @ _). do 3 apply maponpaths. apply homset_property. 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.