Proposition indexed_cat_to_disp_cat_to_disp_iso_inv_left : f · indexed_cat_id Φ x xx₂ · # (Φ $ identity x) indexed_cat_to_disp_cat_to_disp_iso_inv · indexed_cat_comp Φ (identity x) (identity x) xx₁ = indexed_cat_id Φ x xx₁ · idtoiso (maponpaths (λ h, (Φ $ h) xx₁) (!(id_left _))). Proof. unfold indexed_cat_to_disp_cat_to_disp_iso_inv. rewrite functor_comp. rewrite !assoc'. etrans. { do 3 apply maponpaths. apply (indexed_cat_lunitor_alt Φ). } etrans. { apply maponpaths. rewrite !assoc. apply maponpaths_2. refine (!_). apply (nat_trans_ax (indexed_cat_id Φ x)). } cbn. rewrite !assoc. etrans. { do 2 apply maponpaths_2. apply (z_iso_inv_after_z_iso f_iso). } refine (maponpaths (λ z, z · _) (id_left _) @ _). do 4 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.