Definition make_isPreAdditive (PA : categoryWithAbgrops) (H1 : ∏ (x y z : PA) (f : x --> y), ismonoidfun (to_premor z f)) (H2 : ∏ (x y z : PA) (f : y --> z), ismonoidfun (to_postmor x f)) : isPreAdditive PA. Proof. exact (H1,,H2). Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.Algebra.BinaryOperations. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.CategoriesWithBinOps. Require Import UniMath.CategoryTheory.PrecategoriesWithAbgrops. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Limits.Zero. Require Import UniMath.CategoryTheory.Limits.Kernels. Require Import UniMath.CategoryTheory.Limits.Cokernels. rewrite <- (id_left (ZeroArrowFrom y)). set (Y := to_postmor_unel A Z (@ZeroArrowFrom A Z y)). + exact (to_quot_mor (@ZeroArrowFrom PA Z a)). apply ArrowsFromZero.