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I was looking for a characteristics of faithful functors without involving setS and then tried to see if they are just monics in Cat.

Let $F: A\rightarrow B$ be a faithful functor, and $G_1,G_2:T \rightarrow A$. Then if $FG_1$ and $FG_2$ are naturally isomorphic then so are $G_1$ and $G_1$. What if $F$ is monic in Cat? Does that imply its faithfulness?

I guess it is not true because I can not find a proof. Thank you very much.

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What's a Cat? I assume it's not feline.. –  nbubis Feb 5 '13 at 14:14
    
@nbubis I mean Category of Cats (Something as cute:D ) –  Hooman Feb 5 '13 at 14:17
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I don't think it's true that a faithful functor is even monic. For example, any map of sets regarded as a functor is faithful. –  Zhen Lin Feb 5 '13 at 17:32

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up vote 2 down vote accepted

If $F$ is a fully faithful functor, then any isomorphism $F G_1 \cong F G_2$ is induced by a unique isomorphism $G_1 \cong G_2$. In other words, $F$ is a $2$-monomorphism (the $2$ refers to the $2$-categorical structure of $\mathsf{Cat}$). The converse is also true for groupoids. In general, $F$ is fully faithful iff every morphism $F G_1 \to F G_2$ is induced by a unique morphism $G_1 \to G_2$, where $G_1,G_2$ are arbitrary functors (whose codomain is the domain of $F$).

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