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.