Positive model theory of interpretations

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 $\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ coherent functors that vanishes precisely on positively closed models, then we study its functorial properties.