What's the name of this operator? Let $f,g$ be functions in $C^A$ and $C^B$ respectively.
Let $\boxtimes:C^A \times C^B \to (C\times C)^{A \times B}$ s.t. 
$f\boxtimes g(a,b)=(f(a),g(b))$
It seems not the tensor product, nor Cartesian product. Then can we call it direct product? But it seems the term 'direct product' often used on operator between structures.
 A: For the viewpoint of category theory, your map is just $f\times g$ -- it is the image of $f$ and $g$ under the product bifunctor $(-)\times(-)$.
More verbosely, if you compose $f$ and $g$ with the projection maps from $A\times B$, then you get maps $f\circ \pi_1: A\times B \to C$ and $g\circ \pi_2: A\times B \to C$, which factor through $C\times C$ by the universal property of the latter. The mediating morphism is excatly $f\times g$.
If you consider $\mathbf{Set}$ to be a monoidal category by declaring $\times$ to be $\otimes$, then $f\times g$ is indeed the tensor product $f\otimes g$.

On the other hand, in ordinary set theory, we usually identify a function with its graph: $$f=\{\langle x,f(x)\rangle\mid f(x)\text{ is defined}\},$$ and in that sense your $f\boxtimes g$ is of course not the cartesian product of $f$ and $g$. It is closely related though:
$$ f\boxtimes g = \{ \langle\langle a,b\rangle,\langle c,d\rangle\rangle \mid \langle\langle a,c\rangle,\langle b,d\rangle\rangle \in f\times g\}$$
which could be seen as a more vivid justification for Hurkyl's "transpose" terminology.
A: Your map is some transpose of the Cartesian product.
Given a function $f : A \to B$ and another function $g:C \to D$, their Cartesian product is
$$ f \times g : A \times C \to B \times D : (a,c) \mapsto (f(a), g(c)) $$
The notion of "transpose" enters the picture in the sense that you are switching from thinking about a function $A \to C$ to thinking of an element of $C^A$.
