Lemma EqualizerEndo_is_identity {y z : C} {f g : y --> z} {E : Equalizer f g} (φ : C⟦E, E⟧) (H : φ · (EqualizerArrow E) = EqualizerArrow E) : identity E = φ. Proof. set (H1 := tpair ((fun φ' : C⟦E, E⟧ => φ' · _ = _)) φ H). assert (H2 : identity_is_EqualizerIn E = H1). - apply proofirrelevancecontr. apply (isEqualizer_Equalizer E). apply EqualizerEqAr. - apply (base_paths _ _ H2). 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.