Let ${\bf C}$ be a small category. As is well known, the category of presheaves ${\bf Set^C}$ is cartesian closed: it can be a model of intuitionistic propositional logic.
Can it also be a model of classical propositional logic?
That is, does ${\bf Set^C}$ always have an arrow $\bot^{(\bot^A)} \rightarrow A$ for each object $A$? (where $\bot$ denotes an initial object of ${\bf Set^C}$)