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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.

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    Re: alternate ways of defining faithfulness, check magma's answer to http://math.stackexchange.com/questions/166281/why-is-full-faithful-functor-defined-in-terms-of-set-properties2012-12-29

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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.