TITLE: Can the opposite of an elementary topos be an elementary topos?
QUESTION [15 upvotes]: This question is not really about elementary topoi, it is much more about a category $(\mathcal{E}, \Omega)$ admitting a subobject classifier, or about a category with power objects, you can choose the context that inspires you the most. Of course, elementary topoi are the go-to example.

Q: Can we prove, or provide a counter-example, that the opposite of such a category cannot have a subobject classifier, or have power objects, or be an elementary topos?

Rem. It is well known that the opposite of a Grothendieck topos cannot be a Grothendieck topos, but the proof relies on the fact that the opposite of a locally presentable category cannot be locally presentable. This completely obscures the importance of $\Omega$, which is what I would like to know about.
Rem. Even in the most trivial case, why do we know that $\mathsf{Fin}^\circ$, or $(\mathsf{Fin}^C)^\circ$ does not have a subobject classifier?
Clarification. The title mentions elementary topoi because "category with a subobject classifier" would not sound as good.

REPLY [24 votes]: The answer to the question in the title is no, assuming you want to exclude the trivial case of the terminal category.
Let $E$ be an (elementary) topos whose opposite is also a topos. The initial object of a topos is strictly initial (any map into it is an iso), so the terminal object $1$ of $E$ is strictly terminal. Since there is a map from $1$ to the subobject classifier $\Omega$ of $E$, this implies that $\Omega$ is terminal. Hence every object of $E$ has precisely one subobject. In particular, $1$ has only one subobject. But the initial object $0$ of $E$ is a subobject of $1$. So $0 \cong 1$ in $E$, so $E$ is trivial.