Does anyone know where there's a proof of a relativized version of the well-known fact that the topos of sheaves over a locale classifies the points of the locale? What I imagine is something like the following: Let E be a topos, and A a frame in it. Let E' be the topos of internal (in E in some suitable sense) sheaves over the locale of A. Then for any topos F there is an equivalence between - * the category of geometric morphisms from F to E' and * the category whose objects are pairs (f, x) where f: F -> E is a geometric morphism and x: A -> f_*(Omega_F) is a frame homomorphism (with suitable morphisms between these pairs). (Presumably this could be deduced from a more general theorem that generalizes Diaconescu by dealing with toposes of sheaves over sites, not just presheaves over categories.) I feel a bit stupid, because I know topos theorists take results like this for granted all the time. But I can't track down any account of the details, nor quite convince myself that methods such as Joyal and Tierney's ("pretend E is ordinary sets but take care to reason constructively") do the trick in this case. Steve Vickers. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++