That's half of it. If you have another functor going the other way, with some properties on how the two functors relate, then you get what's called an equivalen

That's half of it. If you have another functor going the other way, with some properties on how the two functors relate, then you get what's called an equivalence of categories. [1] Basically, you need both ways to compose the two functors to be "equivalent" to the identity functor in each category (equivalent as shown by a reversible natural transformation). [1]: http://en.wikipedia.org/wiki/Equivalence_of_categories http://en.wikipedia.org/wiki/Equivalence_of_categories