Poset-enriched pretoposes and compact ordered spaces

We provide a characterisation of the category $\mathsf{K}\mathsf{O}\mathsf{r}\mathsf{d}$ of Nachbin’s compact ordered spaces as a poset-enriched category. Up to equivalence, $\mathsf{K}\mathsf{O}\mathsf{r}\mathsf{d}$ is the only non-degenerate poset-enriched pretopos whose terminal object is a (discrete) generator and in which every object is covered by an order-filtral object. Order-filtral objects satisfy an appropriate form of compactness and separation. Throughout, we make extensive use of the internal language of poset-enriched pretoposes.