Proposition is_cartesian_indexed_cat_factorisation_commutes : is_cartesian_indexed_cat_factorisation · # (Φ $ g) ff · indexed_cat_comp Φ f g yy = hh. Proof. unfold is_cartesian_indexed_cat_factorisation. rewrite !assoc'. etrans. { do 2 apply maponpaths. rewrite !assoc. apply maponpaths_2. rewrite <- functor_comp. rewrite z_iso_after_z_iso_inv. rewrite functor_id. apply idpath. } rewrite id_left. rewrite z_iso_after_z_iso_inv. apply id_right. 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.