It is elementary that an injective map from a finite set to itself is surjective, and from the rank-nullity theorem, an injective map from a finite dimensional vector space to itself is also surjective. Further, by the Ax-Grothendieck theorem, an injective polynomial map $p:\mathbb C^n \to \mathbb C^n$ is surjective.
Is there any sort of categorical framework for this? Any characterization or sufficient condition where monic endomorphisms are epic? Perhaps something along the lines of "If a category satisfies property $P$, then the compact objects have all self-monics are epic." Baring this, are there any other interesting or useful examples where the property holds?