\begin{definition}[Definition:Coproduct/Sets]
Let $S_1$ and $S_2$ be sets.
A '''coproduct''' $\struct {C, i_1, i_2}$ of $S_1$ and $S_2$ comprises a set $C$ together with mappings $i_1: S_1 \to C$, $i_2: S_2 \to C$ such that:
:for all sets $X$ and mappings $f_1: S_1 \to X$ and $f_2: S_2 \to X$:
::there exists a unique mapping $h: C \to X$ such that:
:::$h \circ i_1 = f_1$
:::$h \circ i_2 = f_2$
Hence:
::$\begin{xy} \xymatrix@L+2mu@+1em{
& C \ar@{-->}[dd]_*{h} & \\
S_1 \ar[ru]^*{i_1} \ar[rd]_*{f_1} & & S_2 \ar[lu]_*{i_2} \ar[ld]^*{f_2} \\
& X &
}\end{xy}$
:is a commutative diagram.
\end{definition}