TITLE: Left/right exact functor "in nature" which is not a right/left adjoint
QUESTION [18 upvotes]: Let $\mathcal{F}$ and $\mathcal{G}$ be abelian categories.  It is well-known that if a functor $\phi : \mathcal{F} \rightarrow \mathcal{G}$ has a right-adjoint (so $\phi$ is itself a left-adjoint to some other functor), then $\phi$ is right exact.  Similarly, if $\phi$ has a left-adjoint (so $\phi$ is itself a right-adjoint to some other functor), then $\phi$ is left-exact.
I was going through the list of left- and right-exact functors I know, and they all were covered by the above condition.  Can someone give me examples that appear "in nature" (so aren't too artificial) of left- or right-exact functors that are not adjoints?
I'm aware of the adjoint functor theorem, but its conditions are much stronger than just being left- or right-exact.

REPLY [2 votes]: Inverse image of sheaves is exact, but generally only has a right adjoint (direct image). 
Also, for locally compact Hausdorff spaces, direct image with proper supports is left exact, but does not have any adjoints in general unless we pass to the derived category.