Classifying topos of strict partial orders ($[\mathbf{FinSPos},\mathbf{Set}]$)

Let $\mathbf{FinSPos}$ contain finite strict partial orders and strict order-preserving functions. Its covariant set-valued functor category classifies the strict theory. A finite presentation with a strict cycle is contradictory; every other finite diagram gives a finite strict partial order. Equalizing finitely many element identifications either gives such an order or forces a forbidden cycle, so the internal model-to-flat-functor construction remains valid. The reflexification of a strict partial order induces a surjective geometric morphism to the weak-order classifier, but not a geometric embedding because strict maps exclude collapse of comparable elements.