Adjoints with parameters
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.