I'm trying to show that the left adjoint of forgetful functor from abelian groups to groups is the abelianization. After a while of thinking, I think it amounts to showing that the abelinization of a group $G$ satiefies the universal property that every map from $G$ to an abelian group $H_{\mathrm{ab}}$ factor through the quotient of $G$ mod the commutators.
Is that sufficient? Thanks for any comment.