Definition:Coproduct/Sets

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} \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.