Let $(X,O_{X})$ be a ringed space and $\mathcal{F^{\bullet}}$ be a bounded below complex of $O_{X}$-modules on $(X,O_{X})$. On [Stacks Project 20.25.1][1] it is shown that there is a weakly convergent spectral sequence: $$E_{2}^{p,q}=H^{p}(Tot(\mathcal{C}(\mathfrak{U},\underline{H}^{q}(\mathcal{F^{\bullet}})))$$

converging to the hypercohomology $\mathbb{H}^{p+q}(X,\mathcal{F}^{\bullet})$. I was looking for a reference in which this is done on ringed sites instead of ringed spaces. For example, on the étale site over some smooth variety. One would assume that this extends to those cases without further problem, but I can't seem to find a reference to quote it from. [1]: https://stacks.math.columbia.edu/tag/08BN