Let $X$ and $Y$ be topological spaces and $F:Sh(X)\to Sh(Y)$, $G:Sh(Y)\to Sh(X)$ be functors between the categories of sheaves over the respective topological spaces. It seems like a very important part of understanding many functors is to know if it has an adjoint. At the same time, a lot of important properties about sheaves can be translated to the level of stalks.
For this reason I assume that if $F$ is left-adjoint to $G$, then there must be some interesting thing going on at the level of stalks. I can't find any discussion about anything like this in Hartshorne. I realize that the chosen categories can be relaxed a bit, since e.g. the sheafification functor is left-adjoint to the forgetful functor, but I'm not sure how ask the right question.
Could anyone elaborate on any of this, i.e. on what the right question to ask is and what the answer is? If there is a nice relationship, I would be very interested to know where this typically fits in as a proof technique.
A possibly partial answer: the stalk functor $()_x:\mathcal F\mapsto \mathcal F_x$ is left adjoint to the skyscraper sheaf functor $S_x$. Thus if $F$ is left adjoint to $G$, then $(F\cdot)_x$ is left adjoint to $GS_x$. This might give you a way of showing $F$ is not left adjoint to $G$, though of course it's unlikely to help you show when $F$ is a left adjoint.