to an object 
.<\/li>\n
to an arrow 
, where:\n- \n
-
. (Identity is preserved by the functor mapping of morphisms.)<\/em><\/li>\n
. (Associativity is preserved by the Functor mapping of morphisms.)<\/em><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
That’s the standard textbook gunk for defining a functor. But if you look back at the original definition of a category, you should notice that this looks familiar. In fact, it’s almost identical to the definition of the necessary properties of arrows!<\/p>\n
We can make functors much easier to understand by talking about them in the language of categories themselves. Functors are arrows between a category of categories.<\/p>\n
There’s a kind of category, called a small<\/em> category. (I happen to dislike the term “small” category; I’d prefer something like “well-behaved”.) A small category is a category C where Obj(C) and Mor(C) are sets, not proper classes. Alas, more definitions; in set theory, a class<\/em> is a collection of sets that can be defined by a non-paradoxical property that all of its members share. Some classes are sets of sets; some classes are not<\/em> sets; they lack some of the required properties of sets – but still, the class is a collection with a well-defined, non-paradoxical, unambiguous property. If a class isn’t a set of sets, but just a collection that isn’t a set, then it’s called a proper class<\/em>.<\/p>\n
Any category whose collections of objects and arrows are sets, not proper classes, are called small categories. Small categories are, basically, categories that are well-behaved – meaning that their collections of objects and arrows don’t have any of the obnoxious properties that would prevent them from being sets.<\/p>\n
The small categories are, quite beautifully, the objects of a category called Cat<\/b>. (For some reason, category theorists like three-letter labels.) The arrows of Cat<\/b> are the functors: Functors are morphisms between small categories. Once you wrap you head around that, then the meaning of a functor, and the meaning of a structure-preserving transformation become extremely easy to understand.<\/p>\n
Functors come up over and over again, all over mathematics. They’re an amazingly useful notion. I was looking for a list of examples of things that you can describe using functors, and found a really wonderful list on wikipedia.<\/a>. I highly recommend following that link and taking a look at the list. I’ll just mention one particularly interesting example: groups and group actions.<\/p>\n