Lemma 84.20.1. Let $\mathcal{C}$ be a site with fibre products and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Let $K$ be a hypercovering of $X$. With notation as above

$a^* : \textit{Mod}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O})$

is fully faithful with essential image the cartesian $\mathcal{O}$-modules. The functor $a_*$ provides the quasi-inverse.

Proof. Via Remarks 84.15.7 and 84.16.6 and the discussion in the introduction to this section this follows from Lemma 84.18.1. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).