0-truncation of infinity-presheaves Let $\mathcal{C}$ be a small 1-category endowed with a Grothendieck topology. Is it true that the 0-truncation $\tau_0(\mathrm{Shv}_\infty(N(\mathcal{C})))$ of the $\infty$-topos of $\infty$-sheaves on $\mathcal{C}$ is equivalent to the nerve of the ordinary topos of (Set-valued) sheaves over $\mathcal{C}$? According to higher topos theory 6.4.1.3, the 0-truncation of the $\infty$-topos is the nerve of an ordinary topos, but I can't figure out how to prove that it is the obvious one in my case.
At least for the case where $\mathcal{C}$ has the trivial topology, I tried using the universal property of the 0-truncation wrt. left adjoints to 1-categories (HTT 5.5.6.22), but the problem is that a functor $\mathcal{C} \times \mathcal{D} \rightarrow \mathcal{E}$ need not be a right adjoint even if its transpose $\mathcal{C} \rightarrow \mathcal{E}^\mathcal{D}$ is.
 A: Let my try to give an answer, although I should note that I am not an expert so there is no guarantee that my answer will be correct.
I claim that it suffices to show that $\tau_{\leq 0}\mathcal{P}(\mathcal C)\simeq \operatorname{Fun}(\mathcal C^{\mathrm{op}},\tau_{\leq 0}\mathcal S)$ holds (I am supressing the nerve functor here). Indeed, your category $\mathcal X=\operatorname{Shv}_{\infty}(\mathcal C)$ sits as a left-exact accessible localization inside $\mathcal P(\mathcal C)$, and $\tau_{\leq 0}$ induces a left-exact accessible localization of $0$-truncations (this follows from [HTT, 5.5.6.28]). So if $\tau_{\leq 0}\mathcal{P}(\mathcal C)\simeq \operatorname{Fun}(\mathcal C^{\mathrm{op}},\tau_{\leq 0}\mathcal S)$ holds true, then $\tau_{\leq 0}\mathcal X$ is equivalent to a $1$-category that arises as a left-exact localization of $\operatorname{Fun}(\mathcal C^{\mathrm{op}},\tau_{\leq 0}\mathcal S)\simeq \operatorname{Fun}(\mathcal C^{\mathrm{op}},\mathsf{Set})$ and is therefore a $1$-category of sheaves.
Now the equivalence $\tau_{\leq 0}\mathcal{P}(\mathcal C)\simeq \operatorname{Fun}(\mathcal C^{\mathrm{op}},\tau_{\leq 0}\mathcal S)$  can be seen as follows: The left-hand side is the full subcategory of $\mathcal P(\mathcal C)$ spanned by $0$-truncated presheaves, and a presheaf $F$ is $0$-truncated precisely if $\operatorname{map}(-,F)$ takes values in $\tau_{\leq 0}\mathcal S$. Plugging in the presheaf represented by objects $c\in\mathcal C$ tells us that $F$ must factor through $\tau_{\leq 0}\mathcal S$ (using the Yoneda lemma). But any presheaf $F$ that takes values in $\tau_{\leq 0}\mathcal S$ must already be $0$-truncated: This follows from the observation that any presheaf $G$ can be written as a colimit of representables and $\operatorname{map}(-, F)$ takes colimits to limits in $\mathcal S$, together with the fact that the full subcategory $\tau_{\leq 0}\mathcal S\subset \mathcal S$ is stable under arbitrary limits (HTT, 5.5.6.5).
