Existence of right adjoint for fully faithful functor. Let $F:C \to D$ be a fully faithful functor. I was just wondering if there are adjoint functor type theorems for $F$, by which I mean when can we guarantee $F$ has a right adjoint? 
Is there a similar theorem assuming $C$ and $D$ are abelian categories? Thank you in advance.
 A: If $D$ is cocomplete, then such a $C$ must certainly be closed under colimits in $D$. In many situations, this is sufficient. For instance, if $C$ and $D$ are locally presentable, then it is well known that any cocontinuous functor admits a right adjoint. If $C$ is merely a total category, for instance, if it's locally small and well-copowered with a generator, then again as long as $D$ is locally small, cocontinuity implies the existence of a right adjoint. 
Neither of the above conditions relies at all on the full faithfulness of $F$. If $D$ is locally co-presentable, though, we can get such a condition: $C$ must be closed under $\kappa$-cofiltered limits for some cardinal $\kappa$, as well as under colimits. This is the dual of Theorem 2.48 of Adamek and Rosicky's book on locally presentable categories. 
Under the large-cardinal axiom Vopenka's principle, we can do better: the closure under $\kappa$-cofiltered limits is redundant, and one has that every full subcategory of a locally co-presentable category closed under colimits is coreflective. Vopenka's principle is also equivalent to the statement that a full subcategory of a locally presentable category closed under colimits is coreflective. What this means in practice is that you will never see an example of a colimit-closed full subcategory of a locally presentable or co-locally presentable category which is not coreflective.
