proposition: Lemma PushoutEndo_is_identity {a b c : C} {f : a --> b} {g : a --> c} (Pb : Pushout f g) (k : Pb --> Pb) (kH1 : PushoutIn1 Pb · k = PushoutIn1 Pb) (kH2 : PushoutIn2 Pb · k = PushoutIn2 Pb) : identity Pb = k.
proof: Proof. set (H1 := tpair ((fun hk : Pb --> Pb => dirprod (_ · hk = _)(_ · hk = _))) k (make_dirprod kH1 kH2)). assert (H2 : identity_is_Pushout_input Pb = H1). - apply proofirrelevancecontr. apply (isPushout_Pushout Pb). apply PushoutSqrCommutes. - apply (base_paths _ _ H2). Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.TransportMorphisms UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Limits.Initial UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Coequalizers UniMath.CategoryTheory.Epis UniMath.CategoryTheory.Core.Functors