Lemma isEqualizer_path {hs : has_homsets C} {x y z : C} {f g : y --> z} {e : x --> y} {H H' : e · f = e · g} (iC : isEqualizer f g e H) : isEqualizer f g e H'. Proof. use make_isEqualizer. intros w0 h H'0. use unique_exists. - exact (pr1 (pr1 (iC w0 h H'0))). - exact (pr2 (pr1 (iC w0 h H'0))). - intros y0. apply hs. - intros y0 X. exact (base_paths _ _ (pr2 (iC w0 h H'0) (tpair _ y0 X))). Defined.
Require Import UniMath.Foundations.All. 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.Monics. Require Import UniMath.CategoryTheory.Retracts.