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I have added to the Examples at structured (infinity,1)-topos a section Canonical structure sheaves on objects in a big topos.
For the moment this only contains the observation that for $\mathbf{H} = Sh(\mathcal{G})$ the big topos on a geometry $\mathcal{G}$, for every object $X \in \mathbf{H}$ its little topos $\mathbf{H}/X$ is canonically equipped with a $\mathcal{G}$-structure sheaf.
This is evident from the discussion at etale geometric morphism, but it nevertheless seems to be noteworthy.
I have added also an inducation on how this canonical structure sheaf is indeed that of $\mathcal{G}$-valued functions on $X$. But more details on this would be desireable. But I have to interrupt now.
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