Let $\mathcal{M}$ be the monoid $M$ viewed as a category (single object, arrows are elements of $M$) and similarly for $\mathcal{N}$ and N.
Let $F : M \to N$, and $G : M \to N$ be two monoid homomorphisms which are also functors from $\mathcal{M} \to \mathcal{N}$. Then a natural transformation from $F \implies G$ is an element $\psi_N \in N$ such that $\psi_N \cdot F(a) = G(a) \cdot \psi_N, \ \forall a \in M$ where $\cdot$ is the monoid law of $N$.
Then what monoid-theoretic concept does this natural transformation correspond to? I've never seen it before.