I'm looking for examples of (non-degenerate) categories $\mathcal{C}$ such that both $\mathcal{C}$ and $\mathcal{C}^{op}$ are toposes (assuming that such categories even exist).
On a related note, I'm looking for examples of (non-degenerate) categories $\mathcal{C}$ such that $\mathcal{C}$ is a topos and $\mathcal{C} \simeq \mathcal{C}^{op}$ (again, assuming that such categories even exist).
EDIT: I'm also looking for examples of (non-degenerate) categories $\mathcal{C}$ such that $\mathcal{C}^{op}$ is a topos.
Thanks!
If $C^{op}$ is a topos then $(-) \sqcup X$ preserves all limits for any object $X$, and in particular if $1$ denotes the terminal object then $1 \sqcup X = 1$. It follows that
$$\text{Hom}(1, Y) \times \text{Hom}(X, Y) \cong \text{Hom}(1, Y)$$
and hence that if $\text{Hom}(1, Y)$ is non-empty then $\text{Hom}(X, Y)$ contains a unique morphism for every $X$; equivalently, $Y$ is terminal.
If $C$ is a topos then it has a subobject classifier $\Omega$, and taking $Y = \Omega$ above we know that $\text{Hom}(1, \Omega)$ is non-empty since $1$ has at least one subobject, namely itself. It follows that $\Omega$ is terminal, so every object $X$ has exactly one subobject.
But every object a priori has at least two subobjects, namely itself and the initial object. So now it follows that every object is the initial object, and hence $C$ is contractible (equivalent to the terminal category).