Diaconescu equivalence for geometric morphisms ($\mathrm{Geom}(\mathcal{E},\mathrm{Sh}(\mathcal{C},J))\simeq {\mathrm{Flat}}_J(\mathcal{C},\mathcal{E})$)

Geometric morphisms into a sheaf topos correspond to continuous flat functors from its site into the domain topos, with continuity meaning that covers become jointly epimorphic families. Pulling back sheafified representables gives the functor; the tensor construction gives the inverse image in the other direction. This representation theorem should not be confused with the separately named Diaconescu theorem about choice and excluded middle.