unbounded derived category of a $\infty$-topos 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.
[Edit]
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.
 A: Note: answer corrected thanks to the comments of Dylan Wilson and Marc Hoyois.
Let $X$ be a topological space.  Its derived category $D(X)$ is the derived category of the abelian category $Ab(Sh(X))$ of sheaves of abelian groups on $X$.  As an infinity-category it may be defined as the infinity-category associated to the model category of chain complexes in $Ab(Sh(X))$, where weak equivalences are quasi-isomorphisms (so either the projective or injective model structure).  Let me denote this by $D_\infty(Ab(Sh(X)))$.
On the other hand let $Sh_\infty(X, D_\infty(Ab))$ denote the infinity-category of sheaves on $X$ with coefficients in the infinity-category $D_\infty(Ab)$ (= $\mathfrak{C}$ in your notation).  This is defined as the full subcategory of the infinity-category of presheaves on $Ouv(X)$, the site of open subsets of $X$, satisfying Cech descent.  Let $Sh_\infty^{hyp}(X, D_\infty(Ab))$ be the hypercompletion, i.e. the full sub-$\infty$-category of sheaves satisfying hyperdescent.
A comparison of these two constructions boils down essentially to the fact that a chain complex of sheaves of abelian groups is the same thing as a sheaf of chain complexes of abelian groups.  However there is a subtlety with regards to the descent condition.  It turns out that $D_\infty(Ab(Sh(X)))$ is equivalent to $Sh_\infty^{hyp}(X, D_\infty(Ab))$.  Hence it embeds fully faithfully into $Sh_\infty(X, D_\infty(Ab))$.
