proposition: Proposition is_cartesian_to_iso_indexed_cat_right : is_cartesian_to_iso_indexed_cat_inv · ff = identity ((Φ $ f) yy).
proof: 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.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.OppositeCategory.Core UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.IndexedCategories.IndexedCategory