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:

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