What is the relation between the categories $\mathbb{N}_0$ and $\mathbb{N}'_0$ as follows: Both objects and arrows of $\mathbb{N}'_0$ are the natural numbers and f is an arrow $f:a\to b$ iff $f+a=b$. On the other side, $\mathbb{N}_0$ has the only object $*$ and the natural numbers are the arrows. $\mathbb{N}_0$ will be the standard interpretation of a monoid as a category. My question is: $$\text{What is the relation between $\mathbb{N}_0$ and $\mathbb{N}'_0$}?$$ At first I thought the latter might be the arrow category of the first, but that arrow category is just trivial because there is only one object. After all, the only relation I see a forgetful functor from the latter to the first.
$$\text{Are there more interpretations of a monoid as a category?}$$
Well yes, but there's more than one arrow, and they're what the arrow category is about :)
Objects of the arrow category of a monoid $M$ are the elements of $M$, and a morphism between two elements $m$ and $n$ is a pair of elements $(k, l)$ such that $nk = lm$. Setting $k = e$ and $M = ℕ$, you get that your $ℕ'$ is a subcategory of the arrow category of $ℕ$.
Alternatively, a simpler description is that this is the "coslice" category $*/M$ of objects under the unique object of $M$. If $M$ is (left) cancellative, that category is a preorder, and as Respawned Fluff said, for $M = ℕ$ this is exactly the standard order $≤$ on $ℕ$.