  We prove analogues of model theory results for $\mathcal{C}\to \mathcal{D}$
coherent functors, including variants of the omitting types theorem and some
results on ultraproduct constructions. We introduce a distributive lattice
valued invariant of $\mathcal{C}\to \mathbf{Set}$ coherent functors that
vanishes precisely on positively closed models, then we study its functorial
properties.
