Let $\mathcal C$ and $\mathcal D$ be two categories. According to Mac Lane's' book (p. 38) Categoires for the Working Mathematician, the dual category of $\mathcal C\times \mathcal D$ is isomorphic to the product category $\mathcal C^*\times\mathcal D^*$. However, I think that they are the same category.
It is clear that both have they same class of objects. To show that the classes of arrows are also the same, we consider two objects $A=(C,D)$ and $A'=(C',D')$, where $C$ and $C'$ are objects of $\mathcal C$ and $D$ and $D'$ are objects of $\mathcal D$. Then, if $h$ is an arrow of $A$ to $A'$ in $(\mathcal C\times\mathcal D)^*$, it is a pair of arrows $(f,g)$, where $f:C'\rightarrow C$ in $\mathcal C$ and $g:D'\rightarrow D$ in $\mathcal D$. Then $h$ can be considered as the pair of arrows $(f,g)$, with $f:C\rightarrow C'$ in $\mathcal C^*$ and $g:D\rightarrow D'$ in $\mathcal D^*$. The proof for the converse is similar, starting with a morphism $h$ of $A$ into $A'$ in the category $\mathcal C^*\times\mathcal D^*$.
This proof leads me to the question in the title: Is $(\mathcal C \times\mathcal D )^*$ isomorphic or equal to $ \mathcal C^* \times\mathcal D^* $?