Horn theory

In the positive finite-limit convention, a Horn theory uses atomic finite-conjunction sequents; uniquely witnessed existential extensions remain Cartesian. Its models are the finite-limit-preserving functors out of its Cartesian syntactic category. This convention does not insert arbitrary existential witnesses or falsity as Cartesian constructors. Its classifier is the presheaf classifier of a Horn theory.

Internal Horn models are finite-limit-preserving functors out of the Cartesian syntactic category. Since that category has finite limits, these are exactly flat functors; the presheaf Diaconescu equivalence for geometric morphisms proves the classifying property. Equivalently the classifier is the covariant functor category on finitely presented set-based models.