Semi-additive category

A semi-additive category is enriched in commutative monoids: each hom-set has a zero and addition, and composition is additive in each variable. It also has finite products and coproducts. Every finite product is canonically a coproduct and conversely, producing finite biproducts.

A biproduct $A\oplus B$ is simultaneously a product in a category and a coproduct in a category, with projections $p_i$ and injections $i_i$ satisfying