Coproduct

Remarks

A coproduct is the colimit of the diagram whose shape is the set $I$ with only identity morphisms, and it is a product in the opposite category. The coproduct of the empty family is an initial object.

Example: in the category of abelian groups, the direct sum $A \oplus B$ with $\iota_A(a) = (a, 0)$ and $\iota_B(b) = (0, b)$ is a coproduct. Given homomorphisms $f_A \colon A \to Y$ and $f_B \colon B \to Y$, the map $f(a, b) = f_A(a) + f_B(b)$ is a homomorphism because $Y$ is abelian, it satisfies $f \circ \iota_A = f_A$ and $f \circ \iota_B = f_B$, and it is forced since $(a, b) = (a, 0) + (0, b)$. In the category of sets the coproduct is the disjoint union, and in the category of groups it is the free product.

Non-example: in the category of fields, $\mathbb{F}_2$ and $\mathbb{F}_3$ have no coproduct, since no field receives homomorphisms from both: a field homomorphism preserves the characteristic.