2-natural transformation?
An adjunction of two variables is a straightforward generalization of both:
the internal hom in a biclosed monoidal category and
by extracting the central pattern.
Let $C$, $D$ and $E$ be categories. An adjunction of two variables or two-variable adjunction
consists of bifunctors
together with natural isomorphisms
If $(\otimes, hom_l, hom_r) : C \times D \to E$ is a two-variable adjunction, then so are
and
giving an action of the cyclic group of order 3. This can be made to look more symmetrical by regarding the original two-variable adjunction as a “two-variable left adjunction” $C\times D \to E^{op}$; see Cheng-Gurski-Riehl.
There is a straightforward generalization to an adjunction of $n$ variables, which involves $n+1$ categories and $n+1$ functors. Adjunctions of $n$ variables assemble into a 2-multicategory. They also have a corresponding notion of mates; see Cheng-Gurski-Riehl.
Mark Hovey. Model Categories, volume 63 of Mathematical Surveys and Monographs. American Mathematical Society, 1999. See Chapter 4.
Eugenia Cheng, Nick Gurski, Emily Riehl, “Multivariable adjunctions and mates”, arXiv:1208.4520.