Now that we know how to transform adjoints, we can talk about whole families of adjoints parametrized by some other category. That is, for each object of the parametrizing category we’ll have an adjunction , and for each morphism of we’ll have a transformation of the adjunctions.
Let’s actually approach this from a slightly different angle. Say we have a functor , and that for each the functor has a right adjoint . Then I claim that there is a unique way to make into a functor from to so that the bijection is natural in all three variables. Note that must be contravariant in here to make the composite functors have the same variance in .
If we hold fixed, the bijection is already natural in and . Let’s hold and fixed and see how to make it natural in . The components are already given in the setup, so we can’t change them. What we need are functions and for each arrow .
For naturality to hold, we need . But from what we saw last time this just means that the pair of natural transformations forms a conjugate pair from to . And this lets us define uniquely in terms of , the counit of , and the unit of by using the first of the four listed equalities.
From here, it’s straightforward to show that this definition of how acts on morphisms of makes it functorial in both variables, proving the claim. We can also flip back to the original viewpoint to define an adjunction between categories and parametrized by the category as a functor from to the category of adjunctions between those two categories.