Proposition is_cartesian_to_iso_indexed_cat_right : is_cartesian_to_iso_indexed_cat_inv · ff = identity ((Φ $ f) yy). Proof. unfold is_cartesian_to_iso_indexed_cat_inv, φ. rewrite !assoc'. etrans. { apply maponpaths. exact (!(nat_trans_ax (nat_z_iso_inv (indexed_cat_id_nat_z_iso Φ x)) _ _ ff) : inv_from_z_iso (indexed_cat_id_z_iso Φ xx) · ff = # (Φ $ identity x) ff · inv_from_z_iso (indexed_cat_id_z_iso Φ ((Φ $ f) yy))). } cbn. rewrite !assoc. refine (!_). use z_iso_inv_on_left. rewrite id_left. use (cancel_z_iso _ _ (indexed_cat_comp_z_iso Φ f (identity x) yy)). exact (cartesian_factorisation_commutes Hff (identity _) ζ). 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.