Proposition is_cartesian_to_iso_indexed_cat_left : ff · is_cartesian_to_iso_indexed_cat_inv = identity xx. Proof. unfold is_cartesian_to_iso_indexed_cat_inv, φ. rewrite !assoc. refine (!_). use z_iso_inv_on_left. use (cartesian_factorisation_unique Hff) ; cbn. rewrite !assoc'. etrans. { apply maponpaths. rewrite !assoc. exact (cartesian_factorisation_commutes Hff (identity _) ζ). } unfold ζ. rewrite !assoc. apply maponpaths_2. etrans. { apply (nat_trans_ax (indexed_cat_id Φ x)). } apply maponpaths_2. refine (!_). apply id_left. 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.