Let $(L, \cdot)$ be a conmutative monoid with an identity $e$ and let $n \geq 1$ be a natural number. Show the following function is bijective.
$F: Hom_{Monoid}(\mathbb{N}^n, L) \to L^n$ where $F(\varphi)= (\varphi(e_1),..., \varphi(e_n))$.
Where $\mathbb{N}^n$ is a monoid with the sum on each entry and $e_i=(0,....1_i,....0)$, the element with $1$ in the $i$-entry and zero elsewhere.
So my idea was to show that $F$ is injective and surjective. I managed to prove it is indeed injective, but I´m struggling with showing it is surjective. Any ideas and thoughts will be a appreciated.
