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.
The map $\psi$ cannot possibly be surjective. When $M = \mathbb{N}^+$, what element of $M$ would map to the identity of $\mathbb{N}^+$?
There is a standard isomorphism for these structures. Let $Hom_{mon}(A,B)$ stand for the set of monoid homomorphisms $A \to B$ where $A$ and $B$ are monoids, and let $\mathbb{N}^+ = \{0, 1, 2, 3, \ldots\}$ be the natural numbers as an additive monoid.
$$ \psi : M \to Hom_{mon}(\mathbb{N}^+, M) $$ where $\psi(m) : \mathbb{N}^+ \to M$ is the map defined by $\psi(m)(n) = n\cdot m = \underbrace{m + m + m +\cdots + m}_n$.
It should be fairly straightforward to show both injectivity and surjectivity of this mapping.
Hope this helps!