Editing Additive category (section)

=== Via preadditive categories ===
A category '''C''' is preadditive if all its [[hom-set]]s are [[abelian group]]s and composition of morphisms is [[bilinear map|bilinear]]; in other words, '''C''' is [[enriched category|enriched]] over the [[monoidal category]] of abelian groups.

In a preadditive category, every finitary [[product (category theory)|product]] is necessarily a [[coproduct]], and hence a [[biproduct]], and [[converse (logic)|converse]]ly every finitary coproduct is necessarily a product (this is a consequence of the definition, not a part of it). The empty product, is  a [[Initial_and_terminal_objects|final object]] and  the empty product in the case of an empty diagram, an [[Initial_and_terminal_objects|initial object]]. Both being limits, they are not finite products nor coproducts. 

Thus an additive category is equivalently described as a preadditive category admitting all finitary products and with the null object or a preadditive category admitting all finitary coproducts and with the null object