Lemma pre_comp_with_z_iso_is_inj {C : precategory} {a b b' : C} {f : a --> b} {g : b --> a} (i : is_inverse_in_precat f g) : ∏ (f' g' : b --> b'), f · f' = f · g' -> f' = g'. Proof. intros f' g' H. apply (maponpaths (compose g)) in H. do 2 rewrite assoc in H. rewrite (is_inverse_in_precat2 i) in H. do 2 rewrite id_left in H. exact H. 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.