In HTT(Higher Topos Theory) Remark7.3.1.19, it it sketched that the proper base change theorem for $\infty$-topos implies the usual proper base change theorem in (unbounded) derived category. However, I can't understand its detail.

Let $A=Ch(Ab)$ be the category of chain complexes of abelian groups endowed with a conbitional model structure, and $\mathfrak{C}=N(A^{\circ})$ be the underlying $\infty$-category. Then, for a $\infty$-topos $X$, it is written that the homotopy category of $\mathfrak{C}$-valued sheaves over $X$, $hShv(X,\mathfrak{C})$ is equivalent to the usual unbounded derived category $D(X)$. I don't understand why this holds. Please explain me its detail.

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Thank you very much for many comments. As far as I understand these comments, it seems that the unboudned derived category of a $\infty$-topos $X$ is equivalent to the homotopy category of hypercomplete $\mathfrak{C}$-valued sheaves $hShv^{hyp}(X,\mathfrak{C})$, and especially if $X$ is hypercomplete, equivalent to $hShv(X,\mathfrak{C})$.

However, since I don't understand the concept of hypercompleteness, I still don't know why this equivalence holds. Is this explained in HTT, HA, or other DAG series? I'd like to know the referrences which study this subject.

notclaim (and I think it might be false) that you get Spaltenstein's unbounded derived category this way. In fact, I think the whole point is that you do not, since base change fails for Spaltenstein's category. Maybe the two agree under some conditions. In any case, you do get a copy of bounded below derived category living inside. $\endgroup$