> [!definition] > > Let $\catc$ be a [[Category|category]] and $A, B \in \obj{\catc}$. *A* **coproduct** of $A$ and $B$ is a pair $(S, \iota_A, \iota_B)$, where $\iota_A: A \to S$, $\iota_B: B \to S$ such that for any pair of morphisms $f_A: A \to C$ and $f_B: B \to C$, there exists a unique morphism $f: S \to C$ such that > $ > \begin{matrix} > && C & \\ > &\overset{f_A}{\huge\nearrow} & \huge\uparrow_{f} & \overset{f_B}{\huge\nwarrow} \\ > A & \overset{\iota_A}{\huge{\rightarrow}} & S & \overset{\iota_B}{\huge\leftarrow} & B\\ > \end{matrix} > $ > More generally, let $\seqi{A}$ be a family of objects. Then their coproduct is a pair $(S, \seqi{\iota})$ such that for any family of morphisms $\seqi{f}$ where $f_i: A_i \to C$, there exists a unique morphism $f: S \to C$ such that $f_i = f \circ \iota_i$.