  A common theme in model theory is to introduce constant symbols naming the
elements of a model. We describe the analogous construction in the language of
syntactic categories/ sites.
  As an application we identify Boolean-valued models with
$\mathbf{Set}$-valued regular functors on the syntactic category. 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 Boolean-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 embedding to a product of coherent
functors.
