I asked this question on math.stackexchange with no luck, so I thought I would try here. In order to make mirror symmetry more compatible with homological machinery, I understand it is common to give the derived bounded category on a variety a "DG-enhancement" by keeping around the data of an affine Cech cover. This is the perspective taken in "Dirichlet Branes and Mirror Symmetry," section 8.2.1, page 586.
Fix an affine cover $\mathcal{U}$ on $X$ a nonsingular variety. The objects of $D^b_\infty(X)$ are bounded complexes of locally free sheaves. We define for each $q$ a complex of degree-$q$ morphisms $$ \mathcal{H}om^q(\mathcal{E}^\bullet, \mathcal{F}^\bullet) = \bigoplus_m \mathcal{H}om(\mathcal{E}^m, \mathcal{F}^{m+q}), $$ (sheaf homs are over $\mathcal{O}_X$), which has a natural differential in increasing index. Combining this with the data of the Cech cover gives us a double complex, with Cech boundaries in one direction and the (degree-$q$ to degree-$(q+1)$) boundary in the other. We define the homset in our category to be the total complex of this double complex in the usual way: $$ \operatorname{Hom}^n_{D^b_\infty(X)}(\mathcal{E}^\bullet, \mathcal{F}^\bullet) := \bigoplus_{p+q = n} \check{C}^p(\mathcal{U}, \mathcal{H}om^q(\mathcal{E}^\bullet, \mathcal{F}^\bullet)) $$
My question: the text asserts that if we take the cohomology of this category (i.e. cohomology of each its morphism complexes), we should recover the usual derived category $D^b(X)$. Is this obvious? There is an underlying spectral sequence computation, but I don't see why it simplifies.
