Is it possible to give a 'categorical' criterion for when an arbitrary element $m$ of a monoid $M$ is invertible, by which I mean, a criterion that ideally only refers to monoid homomorphisms and does not explicitly refer to the monoid structure?
For example, it is (obviously) true that an element $m$ of a monoid $M$ is invertible iff its image under any monoid homomorphism is invertible. But here the right-hand criterion still involves the notion of invertibility in a monoid. Is there a right-hand criterion that does not make use of the explicit monoid structure? I'm guessing the answer will be 'no', but I just wanted to be sure.
UPDATE: The application I have in mind is the following: I want to determine, for an arbitrary algebraic/equational theory $T$, whether I can create a suitable definition of an 'invertible element' of a model $M$ of $T$. So if I could determine a criterion for when an element of a monoid is invertible, which could be stated in a 'categorical' or universal algebraic way that could be applied to any algebraic theory, then this would likely give me what I want.