What is an example of an accessible category $\mathcal C$ which is not essentially small, such that $\mathcal C^{op}$ is finitely-accessible?

More generally, what is an example of an accessible category $\mathcal C$ (not essentially small) such that one of the following related conditions holds?

• $\mathcal C^{op}$ continuous (i.e. $\mathcal C$ has cofiltered limits and the colimit functor $Ind(\mathcal C^{op}) \to \mathcal C^{op}$ has a left adjoint);

• $\mathcal C^{op}$ is precontinuous (i.e. $\mathcal C$ has colimits and cofiltered limits and the colimit functor $Ind(\mathcal C^{op}) \to \mathcal C^{op}$ preserves limits);

• $\mathcal C$ has finite colimits and cofiltered limits, and they commute;

• $\mathcal C^{op}$ is finitely accessible.

The closest thing to an example I can think of is the category $Hilb$ of Hilbert spaces and contractive maps, which is a self-dual locally $\aleph_1$-presentable category. But I don't believe that finite limits commute with filtered colimits in $Hilb$.

What is an example of an accessible category $\mathcal C$ which is not essentially small, such that $\mathcal C^{op}$ is finitely-accessible?

More generally, what is an example of an accessible category $\mathcal C$ (not essentially small) such that one of the following related conditions holds?

• $\mathcal C^{op}$ continuous (i.e. $\mathcal C$ has cofiltered limits and the colimit functor $Ind(\mathcal C^{op}) \to \mathcal C^{op}$ has a left adjoint);

• $\mathcal C^{op}$ is precontinuous (i.e. $\mathcal C$ has colimits and cofiltered limits and the colimit functor $Ind(\mathcal C^{op}) \to \mathcal C^{op}$ preserves limits);

• $\mathcal C$ has finite colimits and cofiltered limits, and they commute;

• $\mathcal C^{op}$ is finitely accessible.

The closest thing to an example I can think of is the category $Hilb$ of Hilbert spaces and contractive maps, which is a self-dual $\aleph_1$-accessible category. But I don't believe that finite limits commute with filtered colimits in $Hilb$.