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.
Adjunctions of variables
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.