Given a cartesian closed category $\mathbf C$ (or any closed monoidal category) the covariant part of the internal Hom-functor is defined simply in terms of it being the right adjoint to the product. Now, one can easily define the contravariant part of the Hom-functor in terms of the counit of the adjunction, and it satisfies certain "naturality-ish" equations such as $(f \times \operatorname{id}_X)^A \circ \eta_{A,X} = (B \times X)^f \circ \eta_{B,X}$ for any objects $A,B,X$, and $f : A \to B$ in $\mathbf C$ where $\eta_{A,-}$ and $\eta_{B,-}$ re the units of $A \times - \dashv -^A$ and $B \times - \dashv -^B$, respectively.
One thus has a collection of adjoint functors, indexed by the objects of $\mathbf C$ satisfying certain conditions with respect to morphisms between the indexing objects, I'm thus lead to wonder if this is an instance of some more general notion of "reasonable" collection of adjoints.