Proposition is_cartesian_indexed_cat {x y : C} {f : x --> y} {xx : Φ x} {yy : Φ y} (ff : xx --> (Φ $ f) yy) (Hff : is_z_isomorphism ff) : @is_cartesian _ indexed_cat_to_disp_cat y x f yy xx ff. Proof. intros w g ww hh ; cbn in *. use iscontraprop1. - exact (is_cartesian_indexed_cat_factorisation_unique Hff g hh). - simple refine (_ ,, _). + exact (is_cartesian_indexed_cat_factorisation Hff g hh). + exact (is_cartesian_indexed_cat_factorisation_commutes Hff g hh). Defined.
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.