Lemma right_inverse_of_iso_is_inverse {C : precategory} {c c' : C} (f : c --> c') (g : c' --> c) (H : is_inverse_in_precat f g) (h : c' --> c) (HH : f · h = identity _) : h = g. Proof. refine (!id_left _ @ _). refine (maponpaths (fun z => z · h) (!is_inverse_in_precat2 H) @ _). refine (!assoc _ _ _ @ _). refine (maponpaths (fun z => g · z) HH @ _). apply id_right. Qed.
Require Import UniMath.Foundations.PartA. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories.