TITLE: When does a site give rise to a Hausdorff topos
QUESTION [8 upvotes]: What conditions on a Grothendieck site $\left(C,J\right),$ are equivalent to the diagonal map $$Sh_J\left(C\right) \to Sh_J\left(C\right) \times Sh_J\left(C\right)$$ being a proper map of topoi?

REPLY [3 votes]: I think Johnstone's Elephant gives a site characterisation of proper maps between toposes; so I guess the problem reduces to finding the site corresponding to the product $Sh_J (C) \times Sh_J(C)$ - this is surely known?