Exponentials "commutes" explicitly i have a question about exponentials in a category $\Bbb{A}$. I have to prove that the following holds: $C^{A\times B}\cong(C^A)^B$. Therefore i have to give two arrows. This can be done on an explicit manner (that is the claim). But what are exponentials preclisely and how to prove this isomorphism? Can someone help me??
Thanks
 A: We must first define the notion of exponentials in a category. The usual thing is that we do is to start with the cartesian product, $\times$. For any object, $X$ of our category, we have a functor, $$X\mapsto X\times Y$$ (I am suppressing the morphisms for simplicity). Sometimes this functor has has a right adjoint. This means that we have that having the following isomorphism $$hom(X\times Y,Z)\sim hom(X,Z^Y).$$ is the property that defines the exponential (The $\sim$ should actually be a natural transformation that is an isomoprphism, but again I suppress for simplicity). This tells us that the functor, $$X\mapsto X\times Y$$ is left adjoint to $$Z\mapsto Z^Y.$$ When such adjoint exist, they are gaurenteed to be unique (up to isomorphism).
You may now use this to prove the exponential law you have above. To see this consider the string of isomorphisms, $$hom(W\times A\times B, C)\sim hom(W\times A, C^B)\sim hom(W,(C^B)^A).$$ Compare this with $$hom(W\times A\times B, C)\sim hom(W,C^{A\times B}).$$
