Let’s say we have an adjunction . That is, functors and and a natural isomorphism .
Last time I drew an analogy between equivalences and adjunctions. In the case of an equivalence, we have natural isomorphisms and . This presentation seems oddly asymmetric, and now we’ll see why by moving these structures to the case of an adjunction.
So let’s set like we did to show that an equivalence is an adjunction. The natural isomorphism is now . Now usually this doesn’t give us much, but there’s one of these hom-sets that we know has a morphism in it: if then . Then is an arrow in from to .
We’ll call this arrow . Doing this for every object gives us all the components of a natural transformation . For this, we need to show the naturality condition for each arrow . This is a straightforward calculation:
using the definition of and the naturality of .
This natural isomorphism is called the “unit” of the adjunction . Dually we can set and extract an arrow for each object and assemble them into a natural transformation called the “counit”. If both of these natural transformations are natural isomorphisms, then we have an equivalence.
For a particular example, let’s look at this in the case of the free-monoid functor as the left adjoint to the underlying-set monoid . The unit will give an arrow , which here is just the inclusion of the generators (elements of ) as elements of the underlying set of the free monoid. The counit, on the other hand, will give an arrow . That is, we take all elements of the monoid and use them as generators of a new free monoid — write out “words” where each “letter” is a whole element of . Then to take such a word and send it to an element of , we just take all the letters and multiply them together as elements of . Since we gave a description of last time for this case, it’s instructive to sit down and work through the definitions of and to show that they do indeed give these arrows.