## Adjoints and universality

Now we have the notion of an adjunction, along with its unit and counit. It’s time to start tying them back into universality.

The unit of an adjunction picks out, for each object , an arrow . This arrow is an object in the comma category . And, amazingly enough, it’s an initial object in that category. Given any other object and arrow we need to find an arrow in so that . Since the obvious guess is . Then we can calculate:

where the second equality uses the naturality of and the third uses the “quasi-inverse” condition we discussed yesterday.

So, an adjunction means that for each and every object the component of the unit gives a universal arrow from to . Dually, for every object the component of the counit gives a couniversal arrow from to .

On the other hand, let’s say we start out with a functor and for each an object and an arrow that is universal from to . Then given an arrow we can build an arrow . By the universality of there is then a unique arrow so that . It’s straightforward now to show that and are the object and morphism functions of a functor , and that is a natural transformation.

Now, say we have functors and and a natural transformation so that each is universal from to . Given an arrow , there is (by universality of ) a unique arrow so that . This sets up a bijection defined by . This construction is natural in because is, and it’s natural in because is a functor. And so this data is enough to define an adjunction .

Dually, we can start with a functor and for each an object and an arrow universal from to . Then we can build up into a functor and up into a natural transformation with each component a couniversal arrow. And this is enough to define an adjunction .

And, of course, we know that giving a universal arrow from to is equivalent to giving a representation of the functor , and dually.

So we have quite a long list of ways to specify an adjunction

- Functors and and a natural isomorphism
- Functors and and natural transformations and satisfying and
- A functor and for each an object and a universal arrow
- A functor and for each an object and a couniversal arrow
- A functor and for each a representation of the functor
- A functor and for each a representation of the functor
- Functors and and a natural transformation so that each component is universal from to
- Functors and and a natural transformation so that each component is universal from to

[...] by finding a collection of universal arrows, which (along with the forgetful functor ) is one of the many ways we listed to specify an [...]

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