Lemma z_iso_Equalizer_of_same_map {y z : C} {f g : y --> z} (E : Equalizer f g) (p:f = g) : is_z_isomorphism (EqualizerArrow E). Proof. induction p. use (make_is_z_isomorphism _ (from_Equalizer_to_Equalizer (identity_asEqualizer f) E)). use z_iso_from_Equalizer_to_Equalizer_inverses. 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.