The categorical notion which is dual to product. The coproduct of a family of objects of a category is an object , together with a family of morphisms such that for every object and every family of morphisms there is an unique morphism such that
for all . The coproduct is unique up to isomorphisms.
In the category of sets, the coproduct is the disjoint union , and is the inclusion. In the category of Abelian groups, the coproduct is the group direct sum , and is the injection of the th summand. In the category of groups, the coproduct is the free product of groups.