I've been trying to solve a problem and it all reduces to prove that not all faithfull functors preserves monomorphisms. But I haven't been able to come up with an example. I was thinking about using the fact that $$f\in Hom(C) \text{ is monic}\iff f^{op}\in Hom(C^{op}) \text{ is epic}$$ Then if I have for example $$f \in Hom(\textbf{Set}) \text{ s.t. } f \text{ is monic but not epic} \Rightarrow f^{op}\in Hom(\textbf{Set}^{op}) \text{ is epic but not monic}$$ So if I consider the functor that sends a category into its opposite then that would do the work.
Is my reasoning correct? If not can you help me with an especific example?
A simple counterexample to the claim that all faithful functors preserve mononmorphisms is to consider functors from the category $$\bullet\to\bullet$$ with only one non-identity arrow. The only non-trivial arrow is a monomorphism, but you can faithfully map it onto any non-monomorphism in any category.