proposition: Proposition is_cartesian_to_iso_indexed_cat_left : ff · is_cartesian_to_iso_indexed_cat_inv = identity xx.
proof: 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.
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