Note: for a category , we write for the hom-class of maps from to .

An adjunction is a pair of functors and with a certain relationship. There are many texts expositing adjunctions, but I have not seen one covering what I think of as their fundamental theorem. This is regrettable, since I think many people would appreciate the simplifying effect this result has on the theory of adjunctions in . My goal today is to cover this theorem using coend calculus.

**Adjunctions in a nutshell.** First, an example. Let be the category of sets and let be the category of -modules for a ring . There are functors sending a set to and a functor sending an -module to its underlying set. There is a natural way of going back and forth between maps of -modules and maps of sets . For a map of -modules, , we get a map of sets by first applying (this really does nothing) and then precomposing with a map of sets sending to the element in the underlying set of where and for . The resulting map is . For a map of sets , we get a map where sends to , where the sum is taken in . What we have here is a natural correspondence between hom sets and . To go one way we apply and then precompose with (kind of like correcting for an error, since and are not inverses), and to go the other way we apply and then postcompose with . This hints at the definition of an adjoint:

**Definition:** Let and be functors. We say that is adjoint to , or that and are adjoint, if there is a natural isomorphism between the functors .

The example also hints at what I call the fundamental theorem of adjunctions in the -category :

**Theorem:** Let and be categories and and be functors. Then

(i) and are naturally isomorphic.

(ii) and are naturally isomorphic.

**Proof:** These are dual; we prove the first. For the isomorphism itself, we have a series of natural isomorphisms:

This finishes the proof.

Using coend calculus like that is nice, but we are also interested in an explicit description of the isomorphisms and ! To achieve this description, notice that the following diagram commutes for each and each :

Take a natural transformation . For each , define a natural transformation such that sends to , to . Define a natural transformation where . The composition

gives the following diagram of mappings:

As this holds for each and each , we have .

Next take a natural transformation . For each , define a natural transformation such that . Define a natural transformation such that . The composition

gives the following diagram of mappings:

As this holds for each and each , .

Hence sends a natural transformation to the natural transformation where for , , and , and sends a natural transformation to the natural transformation such that .

This theorem puts us in a place where we can prove other results about adjoints much more easily:

**Theorem:** The following are equivalent:

(i) The functors and are naturally isomorphic.

(ii)(a) (Universal Morphisms Definition) There is a natural transformation such that, for each , each , and each morphism in , there is a unique morphism in such that .

(ii)(b) (Universal Morphisms Definition) There is a natural transformation such that, for each , each , and each in , there is a unique morphism in such that .

(iii) (The Unit Counit Definition) There are natural transformations and such that the following compositions are the identity morphism, for each and .

**Proof:** . Suppose that the functors and are naturally isomorphic, and take a natural isomorphism . Let be the corresponding natural transformation. Take , , and . There is a unique such that , since is a natural isomorphism. But by the fundamental theorem. This shows (ii)(a).

. Suppose that there is a natural transformation such that, for each , each , and each morphism in , there is a unique morphism in such that . Let be the natural transformation corresponding to . Take and . for each , so that, for each , there is a unique such that . This shows (i).

. Suppose that the functors and are naturally isomorphic, and take a natural isomorphism . Let be the corresponding natural transformation. Take , , and . There is a unique such that , since is a natural isomorphism. But by the fundamental theorem. This shows (ii)(b).

. Suppose that there is a natural transformation such that, for each , each , and each morphism in , there is a unique morphism in such that . Let be the natural transformation corresponding to . Take and . for each , so that, for each , there is a unique such that . This shows (i).

. Suppose that the functors and are naturally isomorphic, and take a natural isomorphism . Let be the natural transformation corresponding to and let be the natural transformation corresponding to . For , we have

and

. Suppose holds. Let be the natural transformation induced by , and let be the natural transformation induced by .

Take . Write for the induced natural transformation. The following diagram commutes by the Yoneda lemma:

So . So is the identity natural transformation. Hence $EH$ is the identity natural transformation.

Take . Write for the induced natural transformation. The following diagram commutes by the Yoneda lemma:

So . So is the identity natural tarnsformation. Hence is the identity natural transformation.

It follows that and are inverse natural transformations, and therefore natural isomorphisms. This shows (i).