We can view a monoid $M$ as a category with a single object. However, there is another way to make $M$ into a category. Take the elements of $M$ as objects, and define $\mathrm{Hom}(x,y)$ to be set of all triples $(x,y,a)$ such that $ax=y$. Define composition such that $(y,z,b) \circ (x,y,a) = (x,z,ba)$ and identity arrows by $\mathrm{id}_x = (x,x,1).$
I'd like to learn more about this construction. Does it have a name?
What's going on here is that you've allowed the monoid $M$ to operate on the set $M$, and you've successfully expressed this monoid action as a category.
For a general monoid $M$ acting on a set $X$, you can view the elements of $X$ as objects of a category and use the elements of $M$ as arrows between the objects.