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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 H=Sh(𝒢) the big topos on a geometry 𝒢, for every object X∈H its little topos H/X is canonically equipped with a 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 𝒢-valued functions on X. But more details on this would be desireable. But I have to interrupt now.
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