Biproduct

A finite family $(A_i{)}_{i\in I}$ in a category with zero morphisms has a biproduct when both its coproduct and its product exist and the canonical morphism from the coproduct to the product is an isomorphism (Canonical morphism from a finite coproduct to a finite product).

When this happens, either object may be written

${⨁}_{i\in I}{A}_i,$

since the canonical comparison identifies them uniquely. For two objects one writes $A\oplus B$, and for the empty family one speaks of the empty biproduct.

Transporting the product structure across the canonical comparison to the coproduct object makes that comparison the identity. The resulting common object, injections, and projections form the associated biproduct diagram; this is the convention used whenever both kinds of structure maps are written on one displayed object.