proposition: Lemma CoequalizerEndo_is_identity {y z : C} {f g : y --> z} {E : Coequalizer f g} (φ : C⟦E, E⟧) (H : (CoequalizerArrow E) · φ = CoequalizerArrow E) : identity E = φ.
proof: Proof. set (H1 := tpair ((fun φ' : C⟦E, E⟧ => _ · φ' = _)) φ H). assert (H2 : identity_is_CoequalizerOut E = H1). - apply proofirrelevancecontr. apply (isCoequalizer_Coequalizer E). apply CoequalizerEqAr. - apply (base_paths _ _ H2). Defined.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Epis UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Retracts