TITLE: Is every right adjoint of a surjective functor fully faithful?
QUESTION [5 upvotes]: Let $F\colon C\rightarrow D$ a functor, which is surjective and $G$ a right adjoint of $F$.
Is $G$ always fully faithful? What if $C,D$ are abelian and $F$ is additive?
I know that it is always faithful. But must it be full?

REPLY [5 votes]: Let $F : \mathbf{Set} \to \mathbf{Vect}$ be the free vector space functor. (Choose any base field $k$ you like.) It is essentially surjective on objects, and with some cardinality tricks you can arrange for it to be actually surjective, if you want. But its right adjoint is only faithful, not full.
Along similar lines, the functor $F : \mathbf{Ab} \to \mathbf{Vect}$ that sends an abelian group $A$ to the tensor product $A \otimes_\mathbb{Z} k$ is essentially surjective on objects (and additive), but the right adjoint is not always full. (Take $k = \mathbb{R}$, for example.)