Proposition is_cartesian_indexed_cat_factorisation_unique : isaprop (∑ gg, gg · # (Φ $ g) ff · indexed_cat_comp Φ f g yy = hh). Proof. use invproofirrelevance. intros φ₁ φ₂. use subtypePath. { intro. apply homset_property. } use (cancel_z_iso _ _ (functor_on_z_iso (Φ $ g) ff_iso)). use (cancel_z_iso _ _ (indexed_cat_comp_z_iso Φ f g yy)). exact (pr2 φ₁ @ ! (pr2 φ₂)). 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.