Tensor product of left exact morphisms

Dear categorists, If G is a group, then [G,Set] is the classifying topos for right G-torsors. What about the classifying topos for possibly non-transitive torsors? I'm not very adept at these calculations, but if I construct it as a subtopos of the classifying topos for G^op-sets, it appears to come out as [X,Set] where X is the category of finitely presentable, free, non-empty G^op-sets. Similarly, if G is a groupoid, the corresponding classifying topos appears to be [X,Set], where X is the full subcategory of [G^op, Set] on those finite coproducts of representables which have global support. Is this correct? Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]