maybe look up some examples of it

examples of functors along with proofs of the fact that they are functors, the internet is full if it

I am having problems finding those proofs, so send me some links if you can. I do appreciate all your comments.

Have you seen the proofs that, for functor F:C \to D, the existence of a universal morphism from d \in Ob(D) to F implies the existence of a left adjoint to F? Or that the coend formula for left Kan extensions gives you a functor? I believe you can find both in Mac Lane. That book alone is full of proofs that certain things are functors.

And Jens is right on several counts. First, you've told us the action of F on two objects and one morphism; a functor acts by definition on all morphisms and functors of its domain.

Second, you've tried to send the only morphism you define F on to an object that is not in the category of sets. This violates one of the definitional conditions of F being a functor Set \to Set.

And while "associate" may not imply functionality across the whole of mathematical literature, it does sometimes take that meaning in specific contexts. This is one of them. If you're super hung up on wikipedia's phrasing, look in any one of many category theory texts to disambiguate: they all require the "association" of $f$ to $F(f)$ to be functional.

The big problem is this: imagine--keeping A,B, and f as in the original post--if I purported to give the "functor" F such that F(f)= all finite subsets of the naturals with the topology induced by the down-sets of the ordering given by inclusion. Where would you locate the problem: my functor, or the category of sets?