Adjoint Functors (Pt. III)
Point of Post: This is a continuation of this post.
The Limit and Colimit Functors
As a last example we are going to assume we have a category and a small category for which every diagram has a limit and a colimit. We want to define functors and where is the functor category. We already know how this should act on objects–it just takes a diagram and assigns to it either or . The question then is exactly how to figure out what the direct limit of a morphism should be.
To this end suppose that we have an arrow in . We then have arrows for each an object of . Note then though that we have arrows which by the definition of limits gives us a unique arrow with (where of course, the and are the cone maps associated to and respectively). Of course, it’s easy to see that if we have the composition of two natural transformations then satisfies
and since and agree on (obviously the cone maps ) we may conclude that they are equal. Noticing finally that if then and thus . Thus, we may conclude that, with these definitions, is a functor.
Very similarly we may construct a functor which on objects is obviously and on arrows sends to the unique arrow with where and are the cocone maps.
Ok, so with all of this defined we’d like to discuss a certain series of adjunctions involving these two functors and another one we’ve seen before. Namely, define the diagonal functor by taking each object in to the functor with for all and for all arrows , and taking each arrow to the natural transformation with for all (it’s easy to verify this really is a natural transformation).
In particular, we claim that .
To show that we being by defining the isomorphisms . How do we define such an isomorphism? Well, what exactly is a natural transformation ? Well, it’s a set of maps –and that’s it, since the commutativity is trivial since for all morphisms . Ok, great! Clearly then we can make into a bijection by merely defining to be the unique arrow with . Let’s then verify that these isomorphisms are natural in each entry. To this end let be some natural transformation. We must then verify that . To do this let be some natural transformation we see then that
and since our desired maps agree on they must be equal.
Now, for naturality in the other variable let be an arrow in , we must verify then that . To this end, let be a natural transformation. We check then that
Thus, we see that as claimed.
Of course, dualizing the proof one can show that and thus we may finally conclude that:
Theorem: Let be a category which admits all limits (resp. colimits) for diagrams . Then, the functor (resp. ) is right adjoint (resp. left adjoint).
Uniqueness of Adjoints
As a last point we note that right or left adjoints to a given functor are necessarily unique. Indeed, if and then we see that
for all objects . We see then by Yoneda’s lemma that naturally for all objects . So, . The other cases are similar.
 Mac, Lane Saunders. Categories for the Working Mathematician. New York: Springer-Verlag, 1994. Print.
 Adámek, Jirí, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: the Joy of Cats. New York: John Wiley & Sons, 1990. Print.
 Berrick, A. J., and M. E. Keating. Categories and Modules with K-theory in View. Cambridge, UK: Cambridge UP, 2000. Print.
 Freyd, Peter J. Abelian Categories. New York: Harper & Row, 1964. Print.
 Mitchell, Barry. Theory of Categories. New York: Academic, 1965. Print.
 Herrlich, Horst, and George E. Strecker. Category Theory: An Introduction. Lemgo: Heldermann, 2007. Print.