Proposition indexed_cat_lunitor {x y : C} (f : x --> y) (xx : Φ x) : identity ((Φ $ f) xx) = # (Φ $ f) (indexed_cat_id Φ x xx) · indexed_cat_comp Φ (identity x) f xx · idtoiso (maponpaths (λ g, (Φ $ g) xx) (id_left f)). Proof. exact (pr122 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.