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 , and be categories. An adjunction of two variables or two-variable adjunction
consists of bifunctors
together with natural isomorphisms
If is a two-variable adjunction, then so are
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” ; see Cheng-Gurski-Riehl.
There is a straightforward generalization to an adjunction of variables, which involves categories and functors. Adjunctions of 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.