Category exponent isomorphism The book Categories for Working Mathematician - Mac Lane have a exercise described thus:
For categories $A$, $B$, and $C$ establish natural isomorphisms
$ \displaystyle C^{A \times B} \cong (C^B)^A $.
I`m not hitting the natural isomorphism.
 A: I would start like this: let's try to define the isomorphism on objects first.
So, an object of $C^{A\times B}$ is a functor $f: A\times B \longrightarrow C$. And we have to associate it with a functor $\Phi (f) : A \longrightarrow C^B$.
Which means that, first of all, for each object $a \in A$, I have to find an object $\Phi (f) (a) \in C^B$; that is, a functor $\Phi(f) (a) : B \longrightarrow C$.
That is, for each object $b \in B$, I have to define an object $\Phi (f) (a) (b) \in C$. Right?
So, if I just have the functor $f: A \times B \longrightarrow C$, who could be that $\Phi (f) (a) (b) \in C$?
EDIT. Dear Marcelo, how is it going? Assuming you've found who is $\Phi (f) (a) (b) \in C$, you've just got your isomorphism
$$
\Phi : C^{A\times B} \longrightarrow (C^B)^A
$$
defined on objects. Before talking about defining it on morphisms (btw, who are the morphisms of $C^{A\times B}$?), let's talk about its inverse (just on objects too, for the moment)
$$
\Psi: (C^B)^A \longrightarrow C^{A \times B} \ .
$$
So, we would start with an object $g\in (C^B)^A$ -which means a functor $g: A \longrightarrow C^B$- and we would look for an object $\Psi (g) \in C^{A \times B}$ to associate with. That is, $\Psi (g)$ must be a functor
$$
\Psi (g) : A\times B \longrightarrow C \ .
$$
That is, given an object $(a,b) \in A\times B$, we have to define $\Psi (g) (a,b) \in C$. And the only thing at our disposal is the functor $g: A \longrightarrow C^B$.
Well, after thinking for a while, sure enough everybody finds the only way to play with these data:
$$
\Psi (g) (a,b) = g(a)(b) \ .
$$
Which means: you apply the functor $g(a): B \longrightarrow C$ to the object $b\in B$.
Now it is easy to check that $\Phi \circ \Psi$ and $\Psi \circ \Phi$ are identities on objects (assuming you've already found the definition of $\Phi$).
If you are willing to do it and you have troubles about defining $\Phi$ and $\Psi$ on morphisms, just let me know it.
