Universal Property of Free Abelian Group on Set

Theorem
Let $S$ be a set.

Let $(\Z^{(S)}, \iota)$ be the free abelian group on $S$.

Let $G$ be an abelian group.

Let $f : S \to G$ be a mapping.

Then there exists a unique group homomorphism $g : \Z^{(S)} \to G$ with $g \circ \iota = f$:
 * $\xymatrix{

S \ar[d]_\iota \ar[r]^{\forall f} & G\\ \Z^{(S)} \ar[ru]_{\exists ! g} }$

Also see

 * Universal Property of Free Module on Set