proposition: Lemma MapPlusIdentityToCommSq {M:ExactCategory} {A B:M} (f:A-->B) (C:M) (AC : BinDirectSum A C) (BC : BinDirectSum B C) : f · ι₁ = ι₁ · (directSumMap AC BC f (identity C)).
proof: Proof. apply ToBinDirectSumsEq. - rewrite assoc'. rewrite (to_IdIn1 BC). rewrite id_right. unfold directSumMap. unfold BinDirectSumIndAr. rewrite assoc'. rewrite BinDirectSumPr1Commutes. rewrite assoc. rewrite (to_IdIn1 AC). rewrite id_left. reflexivity. - rewrite assoc'. rewrite (to_Unel1 BC). unfold to_unel. rewrite zeroRight. unfold directSumMap, BinDirectSumIndAr. rewrite assoc'. rewrite BinDirectSumPr2Commutes. rewrite id_right. apply pathsinv0. use (to_Unel1 AC). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.Notations UniMath.MoreFoundations.PartA UniMath.Algebra.BinaryOperations UniMath.Algebra.Monoids UniMath.Algebra.Groups UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.Kernels UniMath.CategoryTheory.Limits.Cokernels UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.Pushouts UniMath.CategoryTheory.Limits.BinDirectSums UniMath.CategoryTheory.Limits.Opp UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Morphisms UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Subcategory.Full UniMath.MoreFoundations.Propositions