  In this paper, we first introduce a technique that we call "Yoneda
representation of flat functors", based on ideas from indexed category theory;
then we provide applications of this technique to the theory of classifying
toposes. Specifically, we obtain results characterizing 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 the topos [C^op, Set].
