hmm, that's a good point. i guess ideally i'd want $K$ to use the morphisms in some way. The entry $K(A, B)$ is usually quantifying some sort of notion of similarity between $A$ and $B$. I'm not sure what constraints to impose in general though

@The User: Many of the other answers were not applications of category theory in this sense. The difficulty in both asking and answering a question along these lines was already addressed comments, as well as in some of the answers.

@Yemon: Definitely, reference to the morphisms is necessary to fully realize the categorical viewpoint. My category theorist friend would be horrified to see me make no mention of what the functor does on morphisms! I hope he can forgive me if he ever reads this.

@Yemon: My comment has been transferred to an answer. Questions like this concerning category theory are always tricky to ask as well as answer. I think the power of category theory comes from its usefulness as an organizational tool, and that it gives an easier, more structured way to look at the big picture. I feel like the best way to see the usefulness of category theory in these terms is by just seeing a lot of examples.