proposition: Lemma AbelianPreCat_op_eq2 {X Y : A} (a b c d : X --> Y) : (Abelian_mor_to_BinProd (Abelian_mor_to_BinProd a b) (Abelian_mor_to_BinProd c d)) · (Abelian_mor_from_to_BinProd (CokernelArrow (CokernelOfDiagonal (to_BinProducts A Y Y))) (CokernelArrow (CokernelOfDiagonal (to_BinProducts A Y Y)))) = Abelian_mor_to_BinProd ((Abelian_mor_to_BinProd a b) · CokernelArrow (CokernelOfDiagonal (to_BinProducts A Y Y))) ((Abelian_mor_to_BinProd c d) · CokernelArrow (CokernelOfDiagonal (to_BinProducts A Y Y))).
proof: Proof. unfold Abelian_mor_to_BinProd. unfold Abelian_mor_from_to_BinProd. use BinProductArrowsEq. - rewrite BinProductPr1Commutes. rewrite <- assoc. rewrite BinProductPr1Commutes. rewrite assoc. rewrite BinProductPr1Commutes. apply idpath. - rewrite BinProductPr2Commutes. rewrite <- assoc. rewrite BinProductPr2Commutes. rewrite assoc. rewrite BinProductPr2Commutes. apply idpath. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.Algebra.Monoids UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Coequalizers UniMath.CategoryTheory.Limits.Kernels UniMath.CategoryTheory.Limits.Cokernels UniMath.CategoryTheory.Limits.Pushouts UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.BinDirectSums UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Abelian