Presheaf classifier of a Horn theory ($\mathbf{Set}[\mathbb{T}]\simeq [({\mathcal{C}}_{\mathbb{T}}^{\mathrm{cart}}{)}^{\mathrm{op}},\mathbf{Set}]$)

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.