Definition epis_Pushout_eq2 {x y z : A} (E1 : Epi A x y) (E2 : Epi A x z) (BinCoprod : BinCoproduct (to_Kernels A x y E1) (to_Kernels A x z E2)) (coker : Cokernel (to_Zero A) (BinCoproductArrow BinCoprod (KernelArrow (to_Kernels A x y E1)) (KernelArrow (to_Kernels A x z E2)))) : (KernelArrow (to_Kernels A x z E2)) · CokernelArrow coker = ZeroArrow (to_Zero A) (to_Kernels A x z E2) coker.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.Limits.Zero. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Limits.Kernels. Require Import UniMath.CategoryTheory.Limits.Cokernels. Require Import UniMath.CategoryTheory.Limits.Pushouts. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Opp. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.CategoriesWithBinOps. Require Import UniMath.CategoryTheory.PrecategoriesWithAbgrops. Require Import UniMath.CategoryTheory.PreAdditive. isKernel (to_Zero A) (ZeroArrowFrom x) M (ArrowsFromZero A (to_Zero A) y (ZeroArrowFrom x · M) (ZeroArrow (to_Zero A) _ _)). make_Kernel (to_Zero A) (ZeroArrowFrom _) M _ (MonicKernelZero_isKernel M). - intros y0 Y. apply ArrowsFromZero.