I think every category $C$ has a dual $C^*$.
But is every category the dual of some category?
That is, for every category $C$ does there exist some category $B$ such that $B^* = C$?
I think the answer yes, and such a category is $C^*$:
Let $C$ be a category.
Since $C$ is a category, it has a dual category $C^*$, which, in particular, is a category.
Since $C^*$ is a category, it has a dual $C^{**}$.
Now the double dual category $C^{**}$ equals $C$, namely $C^{**} = C$. (Why?)
Therefore there exists a category $C^*$ such that $C^{**} = C$.
Therefore there exists a category $B$ such that $B^* = C$.
Is this correct?