Lemma z_iso_from_Equalizer_to_Equalizer_inverses {y z : C} {f g : y --> z} (E E' : Equalizer f g) : is_inverse_in_precat (from_Equalizer_to_Equalizer E E') (from_Equalizer_to_Equalizer E' E). Proof. use make_is_inverse_in_precat. - apply pathsinv0. use EqualizerEndo_is_identity. rewrite <- assoc. unfold from_Equalizer_to_Equalizer. rewrite EqualizerCommutes. rewrite EqualizerCommutes. apply idpath. - apply pathsinv0. use EqualizerEndo_is_identity. rewrite <- assoc. unfold from_Equalizer_to_Equalizer. rewrite EqualizerCommutes. 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.