Lemma left_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 : h · f = identity _) : h = g. Proof. refine (!id_right _ @ _). refine (maponpaths (fun z => h · z) (!is_inverse_in_precat1 H) @ _). refine (assoc _ _ _ @ _). refine (maponpaths (fun z => z · g) HH @ _). apply id_left. 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.