free abelian group functor

The free group functor \(\mathrm{freeGrp}\) is defined

\begin{align*} \mathrm{freeAb}: \mathrm{Set} \rightarrow& \mathrm{Ab} \\ X \mapsto& \mathrm{FreeAb}(X) \\ (f: X \rightarrow Y) \mapsto& (f': \mathrm{FreeAb}(X) \rightarrow \mathrm{FreeAb}(Y)) \end{align*}

where \(f'\) is the group-homomorphism induced by

\begin{align*} f'(x) = f(x) \end{align*}