Correspondence between Abelian Groups and Z-Modules/Isomorphism of Categories

Theorem
Let $\Z$ be the ring of integers.

Let $\mathbf{Ab}$ be the category of abelian groups.

Let $\mathbf{\mathbb Z-Mod}$ be the category of unitary $\Z$-modules.

Then the: are strict inverse functors.
 * forgetful functor $\mathbf{\mathbb Z-Mod} \to \mathbf{Ab}$
 * associated Z-module functor $\mathbf{Ab} \to \mathbf{\mathbb Z-Mod}$

In particular, $\mathbf{Ab}$ and $\mathbf{\mathbb Z-Mod}$ are isomorphic.