$Sh(B)$-valued models of $(\kappa ,\kappa )$-coherent categories

A basic technique in model theory is to name the elements of a model by introducing new constant symbols. We describe the analogous construction in the language of syntactic categories/ sites. As an application we identify $\mathbf{Set}$-valued regular functors on the syntactic category with a certain class of topos-valued models (we will refer to them as "$Sh(B)$-valued models"). For the coherent fragment $L_{\omega \omega }^g \subseteq L_{\omega \omega }$ this was proved by Jacob Lurie, our discussion gives a new proof, together with a generalization to $L_{\kappa \kappa }^g$ when $\kappa$ is weakly compact. We present some further applications: first, a $Sh(B)$-valued completeness theorem for $L_{\kappa \kappa }^g$ ($\kappa$ is weakly compact), second, that $\mathcal{C}\to \mathbf{Set}$ regular functors (on coherent categories with disjoint coproducts) admit an elementary map to a product of coherent functors.