Q:

Does the property of "reflecting monomorphisms" for a functor, implies its faithfulness?

I can see why

If $F$ is faithful, then it reflects monomorphisms.

My question is does the inverse holds? More generally, I would like to ask:

What are the alternative ways of defining the faithfulness of functors (Apart from "..one which its restriction to home sets is bijective)

Thank you very much.

A:

No. Let $C$ be a groupoid. Then any functor $F : C \to D$ reflects monomorphisms (because every morphism in $C$ is a monomorphism) but most of them are not faithful. 

