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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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I don't understand the second question. (That is not the definition of faithfulness.) –  Qiaochu Yuan Dec 29 '12 at 1:35
    
Re: alternate ways of defining faithfulness, check magma's answer to math.stackexchange.com/questions/166281/… –  alancalvitti Dec 29 '12 at 21:36

1 Answer 1

up vote 3 down vote accepted

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.

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