In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products within a given category.

Let ${\displaystyle C}$ be a category and let ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ be objects of ${\displaystyle C.}$ An object is called the coproduct of ${\displaystyle X_{1}}$ and ${\displaystyle X_{2},}$ written ${\displaystyle X_{1}\sqcup X_{2},}$ or ${\displaystyle X_{1}\oplus X_{2},}$ or sometimes simply ${\displaystyle X_{1}+X_{2},}$ if there exist morphisms ${\displaystyle i_{1}:X_{1}\to X_{1}\sqcup X_{2}}$ and ${\displaystyle i_{2}:X_{2}\to X_{1}\sqcup X_{2}}$ that satisfies the following universal property: for any object ${\displaystyle Y}$ and any morphisms ${\displaystyle f_{1}:X_{1}\to Y}$ and ${\displaystyle f_{2}:X_{2}\to Y,}$ there exists a unique morphism ${\displaystyle f:X_{1}\sqcup X_{2}\to Y}$ such that ${\displaystyle f_{1}=f\circ i_{1}}$ and ${\displaystyle f_{2}=f\circ i_{2}.}$ That is, the following diagram commutes:

The unique arrow ${\displaystyle f}$ making this diagram commute may be denoted ${\displaystyle f_{1}\sqcup f_{2},}$ ${\displaystyle f_{1}\oplus f_{2},}$ ${\displaystyle f_{1}+f_{2},}$ or ${\displaystyle \left[f_{1},f_{2}\right].}$ The morphisms ${\displaystyle i_{1}}$ and ${\displaystyle i_{2}}$ are called canonical injections, although they need not be injections or even monic.

The definition of a coproduct can be extended to an arbitrary family of objects indexed by a set ${\displaystyle J.}$ The coproduct of the family ${\displaystyle \left\{X_{j}:j\in J\right\}}$ is an object ${\displaystyle X}$ together with a collection of morphisms ${\displaystyle i_{j}:X_{j}\to X}$ such that, for any object ${\displaystyle Y}$ and any collection of morphisms ${\displaystyle f_{j}:X_{j}\to Y}$ there exists a unique morphism ${\displaystyle f:X\to Y}$ such that ${\displaystyle f_{j}=f\circ i_{j}.}$ That is, the following diagram commutes for each ${\displaystyle j\in J}$:

The coproduct ${\displaystyle X}$ of the family ${\displaystyle \left\{X_{j}\right\}}$ is often denoted ${\displaystyle \coprod _{j\in J}X_{j}}$ or ${\displaystyle \bigoplus _{j\in J}X_{j}.}$

Sometimes the morphism ${\displaystyle f:X\to Y}$ may be denoted ${\displaystyle \coprod _{j\in J}f_{j}}$ to indicate its dependence on the individual ${\displaystyle f_{j}}$s.

The coproduct in the category of sets is simply the disjoint union with the maps i_j being the inclusion maps. Unlike direct products, coproducts in other categories are not all obviously based on the notion for sets, because unions don't behave well with respect to preserving operations (e.g. the union of two groups need not be a group), and so coproducts in different categories can be dramatically different from each other. For example, the coproduct in the category of groups, called the free product, is quite complicated. On the other hand, in the category of abelian groups (and equally for vector spaces), the coproduct, called the direct sum, consists of the elements of the direct product which have only finitely many nonzero terms. (It therefore coincides exactly with the direct product in the case of finitely many factors.)

Given a commutative ring R, the coproduct in the category of commutative R-algebras is the tensor product. In the category of (noncommutative) R-algebras, the coproduct is a quotient of the tensor algebra (see Free product of associative algebras).