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.

Artin, M. Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1991.Joshi, K. D. "Coproducts
and Products." Ch. 8 in Introduction
to General Topology. New Delhi, India: Wiley, pp. 189-216, 1983.Kasch,
F. "Construction of Products and Coproducts." §4.80 in Modules
and Rings. New York: Academic Press, pp. 80-84, 1982.Rowen,
L. "Products and Coproducts." In Ring
Theory, Vol. 1. San Diego, CA: Academic Press, pp. 73-76, 1988.Strooker,
J. R. "Products and Sums." §1.5 in Introduction
to Categories, Homological Algebra and Sheaf Cohomology Cambridge, England:
Cambridge University Press, pp. 14-21, 1978.