# When does a site give rise to a Hausdorff topos

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?

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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?
A completely random guess: is it the coproduct $C \coprod C$? –  David Roberts Apr 4 '12 at 21:38