I'm trying to prove this isomorphism. I defined this function
$ \psi: M \rightarrow Hom(\mathbb{N}^{+}, M) \\ m \mapsto \phi(n) $ where $ \phi(n) = \begin{cases} e_M, & \text{if }n\text{ is even} \\ m, & \text{if }n\text{ is odd} \end{cases} $
$\psi$ is obviously injective, and this shows that $|Hom(\mathbb{N^{+}}, M)| \ge |M|$. I have yet to show surjectivity, I've been told to use right inverse definition of surjectivity but I don't quite understand what to do.
edit- $\phi$ is definitely not a homomorphism, oops.
So the question is how would one define this homomorphism and then prove bijectivity.