# Definition:Coproduct/Sets

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## Definition

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} \[email protected][email protected]+1em{ & C \[email protected]{-->}[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.