Universal Property of Abelianization of Group

Theorem
Let $G$ be a group.

Let $G^{\operatorname {ab} }$ be its abelianization.

Let $\pi : G \to G^{\operatorname {ab} }$ be the quotient group epimorphism.

Let $H$ be an abelian group.

Let $f: G \to H$ be a group homomorphism.

Then there exists a unique group homomorphism $g : G^{\operatorname {ab}} \to H$ such that $g \circ \pi = f$:
 * $\xymatrix {

G \ar[d]_\pi \ar[r]^{\forall f} & H\\ G^{\operatorname {ab} } \ar[ru]_{\exists ! g} }$