I'm trying to do exercise 1 of II.5 in the Mac Lane's book Category theory for the working mathematicians.
Te exercise asks me to find a natural bijection
$$\operatorname{Hom}(A\times B,C) \cong \operatorname{Hom}(A,C^B) $$
for small categories $A,B$ and $C$ (so the hom-sets are thought in the category of small categories $\mathbf{Cat}$.)
I thought to define the following functors
$$
\begin{array}{rcl}
F: \mathbf{Cat}^* \times\mathbf{Cat}^* \times \mathbf{Cat} & \longrightarrow & \mathbf{Set} \\
(A,B,C) & \longmapsto & \operatorname{Hom}(A\times B, C)
\end{array}
$$
and
$$
\begin{array}{rcl}
G: \mathbf{Cat}^* \times\mathbf{Cat}^* \times \mathbf{Cat} & \longrightarrow & \mathbf{Set} \\
(A,B,C) & \longmapsto & \operatorname{Hom}(A,C^B) .
\end{array}
$$
But, as functors, I have to define them also for arrows. And here is my problem. If $H:A\times B\times C^* \rightarrow A'\times B'\times {C'}^*$ is a functor (an arrow in $\mathbf{Cat}\times\mathbf{Cat}\times\mathbf{Cat}^*$) I have no idea of how to define $FH$. If I'm not wrong, it is a map (in the usual set) which takes a functor $T:A\times B \rightarrow C$ and gives a functor $FH(T):A'\times B'\rightarrow C'$. So given $T$ and objects $(a',b')$ I have to define
$$ FH(T)(a',b') $$
and also
$$FH(T)(f',g') $$
but I have no clue about it.
Is my procediment fine? In this case, can you help me, please?
Best Answer
Remember that you don't have to define Hom functors from scratch; you should already know how they work. Given $\langle f,g,h\rangle:\langle A,B,C\rangle\to\langle A',B',C'\rangle$, $F(f,g,h)(a:A\times B\to C)$ is just $a\mapsto h\circ a\circ (f\times g)$. (Notice that you have the variance backwards above--the first argument of a Hom functor is contravariant, the second covariant.) $G$ requires a little more thought, but it's easy to sort out with a little time.
$H$ is, I think, where you're getting confused. All you need to show is that for every $A,B,C$ there's a function between sets $H_{A,B,C}:F(A,B,C)\to G(A,B,C)$ that commutes with the $F(f,g,h)$ and the $G(f,g,h)$. That's it. You don't need any extra functors. These functions are given by the currying operation described above.