Category Theory/Additive categories

biproduct Let be a category, and let be a family of objects in . A biproduct of is an object of that is usually denoted as and for which there exist arrows and for all that have the following properties: and , together with the morphisms , constitutes a coproduct in the category , together with the morphisms , constitutes a product in the category additive category An additive category is a category that satisfies each of the following requirements: Every morphism in has a kernel and a cokernel For every two objects of , there exists a biproduct For every two objects of , the assignment , where is the morphism that arises from postcomposing the morphism (where shall denote the diagonal) with the anti-diagonal , turns into an abelian group