Consider the adjunction $\mathcal{C} \mathrel{\substack{\mathcal{F}\\\rightleftarrows\\ \mathcal{G}}} \mathcal{D} $ together with unit $\eta: I_{\!_{\mathcal{C}}} \rightarrow \mathcal{G} \mathcal{F} $ and counit $\varepsilon:\mathcal{F} \mathcal{G} \rightarrow I_{\!_{\mathcal{D}}}$. Then by theorem in Mac Lane's book, $\mathcal{G}$ is faithful if and only if the counit $\varepsilon_{\!_D}$ is epic for $D \in \mathcal{D}$. Rafael's theorem (this is one part of the theorem, and the other is just its dual) for separable functors states that $\mathcal{G}$ is separable if and only if the counit $\varepsilon$ cosplits. From the definition of separability, we have: if $\mathcal{G}$ is separable, then $\mathcal{G}$ is faithful. I also know that a functor is fully separable iff it is fully faithful, but do we have the following: If $\mathcal{G}$ is faithful, then it is separable? If not, can we get a counter example?
Thanks,
To elaborate on Qiaochu's comment: The theorem written on the nLab states that $\mathcal G$ is separable iff the counit $\epsilon$ has a retraction, i.e. a left inverse. This implies that $\epsilon$ is a monomorphism, but is not equivalent to it.
In case that was your confusion: Qiaochu himself has written a nice post on his blog that I read a few weekss ago explaining the difference between epic and having a right inverse. https://qchu.wordpress.com/2012/10/01/split-epimorphisms-and-split-monomorphisms/