Proposition indexed_cat_runitor {x y : C} (f : x --> y) (xx : Φ x) : identity ((Φ $ f) xx) = indexed_cat_id Φ y ((Φ $ f) xx) · indexed_cat_comp Φ f (identity y) xx · idtoiso (maponpaths (λ g, (Φ $ g) xx) (id_right f)). Proof. exact (pr1 (pr222 Φ) x y f xx). 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.