Yoneda representations of flat functors and classifying toposes

We obtain semantic characterizations, holding for any Grothendieck site (C, J), for the models of a theory classified by a topos of the form Sh(C, J) in terms of the models of a theory classified by a topos [C^op, Set]. These characterizations arise from an appropriate representation of flat functors into Grothendieck toposes based on an application of the Yoneda Lemma in conjunction with ideas from indexed category theory, and turn out to be relevant also in different contexts, in particular for addressing questions in classical Model Theory.