A coproduct is the categorical dual of a category 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 a unique morphism 
such
that
 |
for all 
. The coproduct is unique up to isomorphism.
In the category of sets, the coproduct is the disjoint union 
, and 
is the inclusion
map. 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.
