The forgetful functor from ℤ-modules to additive commutative groups is
an equivalence of categories.
either use this equivalence to transport the monoidal structure from Module ℤ to Ab,
or, having constructed that monoidal structure directly, show this functor is monoidal.
The forgetful functor from ℤ modules to AddCommGroup is full.
The forgetful functor from ℤ modules to AddCommGroup is essentially surjective.