Let’s look at universal arrows a bit more closely. If we have a functor , an object , and a universal arrow , then we can take any arrow and form the composition . The universal condition says that to every arrow corresponds a unique so that . That is, there is a bijection for every .
It’s even better than that, though: these bijections commute with arrows in . That is, if we have then the naturality condition holds for all . This makes the into the components of a natural isomorphism . That is, a universal arrow from to is a representation of the functor .
Conversely, let’s say we have a representable functor , with representation . Yoneda’s lemma tells us that this natural transformation corresponds to a unique element of , which as usual we think of as an arrow . So a representation of a functor is a universal arrow in .
So, representations of functors are the same as universal arrows. Colimits are special kinds of universal arrows, and since a universal arrow is an initial object in an apropriate comma category it is a colimit. Representations are universals are colimits.
If we swap the category for the dual category , we change limits into colimits, couniversals into universals, and contravariant representations into covariant ones. All the reasoning above applies just as well to . Thus, we see that in representations of contravariant functors are couniversals are limits.
There’s actually yet another concept yet to work into this story, but I’m going to take a break from it to flesh out another area of category theory for a while.