Let $C$ be category with finite limits. For a map $X \rightarrow Y$, the map $C/Y \rightarrow C/X$ "pullback with $X \rightarrow Y$" has a left adjoint, the "forgetful functor". Usually forgetful functors are right adjoint, but that is not completely odd; if we had chosen the category $C \backslash Y$ we would get the reverse.
Is it possible to characterize adjunctions arising in this way? What are some properties that this adjunction enjoys?