Let $(\mathcal C, J)$ be a small subcanonical $\infty$-site, and let $Sh_J(\mathcal C)$ be the $\infty$-topos of sheaves thereon. Then $Sh_J(\mathcal C)$ is itself an $\infty$-site with respect to the canonical topology. There is a fully faithful Yoneda embedding $y: Sh_J(C) \to Sh_{can}(Sh_J(C))$.
Question 1: Is this functor an equivalence?
This question is motivated by Charles Rezk's question here. This question was asked by David Carchedi assuming only that $Sh_J(\mathcal C)$ is an $\infty$-topos, not necessarily a sheaf $\infty$-topos. In that generality, I think the answer is no, but I've lost track of where I got that understanding from too.
At various times I've thought I've known the answer to this question, but I find myself a bit confused tracing through some old conversations. Currently it seems to me this functor is fully faithful. Moreover, it seems to me (modulo size issues, which I think could be dealt with in the worst case by just a bit of universe-juggling) that every $F \in Sh_{can}(Sh_J(C))$ is covered by representables $F_i \in Sh_J(\mathcal C)$, and that each $F_i$ is covered by representables $C_{ij} \in \mathcal C$ by subcanonicity (and since any cover in $J$ is a cover in $can$ after applying Yoneda). Since covers are closed under composition, it seems to me that the $C_{ij} \to X$ form a cover of $X$ by objects of $\mathcal C$. This seems a short step from saying that $Sh_{can}(Sh_J(\mathcal C))$ is generated under colimits by $\mathcal C$, which seems quite close to saying that the answer should be "yes".
Question 2: Can some argument along the above lines be pushed to completion, or are there fundamental difficulties with this approach?
