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](https://ncatlab.org/nlab/show/continuous+category) (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](http://www.tac.mta.ca/tac/volumes/11/11/11-11abs.html) (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 $Hilb$.



Can the dual of a finitely-accessible category be accessible?
In **Accessible Categories: The Foundations of Categorical Model Theory** by *Makkai and Paré*, there is the example of a finitely accessible self-dual category. Apparently the example is due to *Isbell*. This is the category of **sets and partial monomorphisms**. The example appears **right after Prop. 3.4.4** and right before 3.4.5.


