proposition: 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: 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.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Retracts