In set theory, and sometimes in category theory, A^B is just another notation for Hom(B, A), and the latter might be given the alternate notation B -> A. And th reason is that for finite sets, computing cardinalities result in the usual power function of natural numbers - same as Church, then.

Hans

Slightly off topic, but the A^B notation for hom-sets also makes the natural isomorphism we call currying expressable as A^(BxC) = (A^B)^C.

Nathan Bloomfield