Lemma EqualizerInComp {y z : C} {f g : y --> z} (E : Equalizer f g) {x x' : C} (h1 : x --> x') (h2 : x' --> y) (H1 : h1 · h2 · f = h1 · h2 · g) (H2 : h2 · f = h2 · g) : EqualizerIn E x (h1 · h2) H1 = h1 · EqualizerIn E x' h2 H2. Proof. use EqualizerInsEq. rewrite EqualizerCommutes. rewrite <- assoc. rewrite EqualizerCommutes. apply idpath. Qed.
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.