An exercise in The Joy of Cats, p. 59, is as follows:
Show that no finite monoid, considered as a category, has a proper reflective subcategory.
The obvious idea is to let $r : \cdot \to \cdot$ be a reflector. Then by assumption every arrow $f$ factors as $f' \circ r$, where $f'$ is in the subcategory. Now if we can show that $r$ is itself in the subcategory, then we win... But this is giving me some trouble.
The theorem is false for infinite monoids (and the second part of this problem, which I've done, gives a counterexample), but I'm not sure how to leverage finiteness without knowing my monoid is cancellative. We can conclude lots of things by pigeonhole, but I'm not sure how to apply them.
Any help is appreciated ^_^