I'm sorry in advance if this question does not belong on this site. I am curious as to what is "really" going on when you have a family of functors indexed by elements in a base category, all of which admit an adjoint. The two motivating examples for my question are internal homs $ -\otimes X \dashv [X,-] $ (when the category is closed monoidal) and the adjunction $ \pi_{-\times X} \dashv \Gamma : \mathbb{C} \underset{{\leftarrow}}{\to} \mathbb{C}_{/X}$, where $\Gamma$ is the object of sections and $\pi$ denotes the product projection onto $X$, and $\mathbb{C}$ is cartesian closed and complete. In both cases, we have an indexed family of functors which all admit adjoints (we could associate each object to the left or the right adjoint). It feels kind of goofy to just say, "for all $X$ this adjunction exists." Is there a more natural way to look at these structures? I was thinking, at least for the space of sections adjunction, that it might involve the functor $\mathbb{C}\to\ ^{\mathbb{C}/}\operatorname{Cat}$ sending each $X$ to the functor $\pi_{-\times X} $ out of $\mathbb{C}$, and then we might have some functor $\mathbb{C} \to \operatorname{Cat}_{/\mathbb{C}}$, which sends each $X$ to the corresponding $\Gamma$ functor, satisfying a universal property encoding the collective adjointness. Is there some well-known construction capturing this idea?

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    $\begingroup$ Internal homs are a "two-variable adjunction": ncatlab.org/nlab/show/two-variable+adjunction $\endgroup$ – Mike Shulman Dec 5 '16 at 21:56
  • $\begingroup$ Even if the terminology is quite surpassed, I suggest you to look at Gray's paper "Closed categories, lax limits and homotopy limits" and in particular Def 1.1 and the characterization of a "THC situation" as a triple of functors ${\bf A}\to \text{LAdj}({\bf C}, {\bf B})$, ${\bf B}\to \text{LAdj}({\bf C}, {\bf A})$, ${\bf C}\to \text{RAdj}({\bf B}, {\bf A})$. $\endgroup$ – Fosco Dec 6 '16 at 8:34
  • $\begingroup$ Look for adjunctions with parameters in Mac Lane's book. It is pretty standard. $\endgroup$ – HeinrichD Dec 6 '16 at 12:20

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