Fully faithful functors on set-theoretic level

Question

Do there exist two small categories $C$, $D$ such that there is a fully faithful functor $C\rightarrow D$ and a functor injective (surjective, bijective) on objects $C\rightarrow D$ but no fully faithful functor injective (surjective, bijective) on objects $C\rightarrow D$? I just started thinking about this in relation to maps of Kaehler manifolds (it might happen that there is a symplectomorphism, a biholomorphism but no isomorphism of Kaehler manifolds).

Answer 1

Sure, consider the disjoint union of two copies of a monoid $M$ mapping to $M\sqcup *$, where $*$ is the terminal category. Then the functor that collapses both copies of $M$ is fully faithful, but any injective-on-objects or surjective-on-objects functor must have $*$ in its image, so that it's not fully faithful.