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I guess I would be fine with this. I remember back when we created this entry, there was some discussion about what the best way to expose this is. I think opinions generally will differ, but it is certainly true that the subfunctor picture is the best one.
Notice that there is closely related material also at sheaf and maybe there would be reason to try to merge and harmonize this more with that at sieve.
This is a good question.
I think indeed this is one of the very few places where there is something like a gap in Lurie's work:
he doesn't have a notion of (oo,1)-site such that one would have the expected equivalence between oo-sheaves on the oo-site and geometric embeddings into the oo-presheaves.
He does discuss in great detail how the structure of a 1-site on an oo-category may correspond to different notions of oo-sheaves: notably for each site there is the oo-topos of oo-sheaves that satisfy descent with respect to all Cech covers in the site. And then there is the hypercompletion of this, of oo-presheaves that satisfy descent also with respect to all hypercovers.
Ideally one would think that there should be a notion of oo-site that encodes the difference between these.
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