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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.

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up vote 3 down vote accepted

The internal hom of a cartesian closed category, or more generally any monoidal closed category, is part of a parametrised adjunction, called the tensor–hom adjunction. The general case is described in [CWM, Ch. IV, §7, Thm 3]. The bijection $$\textrm{Hom}(A \otimes B, C) \cong \textrm{Hom}(A, \textrm{Hom}(B, C))$$ is natural in $A$, $B$ and $C$, but the adjunction counit $$\epsilon_{B, C} : \textrm{Hom}(B, C) \otimes B \to C$$ is natural in $C$ and extranatural in $B$: this is defined in [CWM, Ch. IX, §4].

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There's a pretty good overview in nLab as well: – Tilo Wiklund Jul 23 '12 at 18:51

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