Lemma 96.19.9. Let $f : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. Assume
$\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$,
for every object $x$ of $\mathcal{X}$ there exists a covering $\{ x_ i \to x\} $ in $\mathcal{X}_\tau $ such that each $x_ i$ is isomorphic to $f(u_ i)$ for some object $u_ i$ of $\mathcal{U}$,
the category $\mathcal{U}$ has equalizers, and
the functor $f$ is faithful.
Then there is a first quadrant spectral sequence of $\Gamma (\mathcal{O}_\mathcal {X})$-modules
\[ E_1^{p, q} = H^ q((\mathcal{U}_ p)_\tau , f_ p^*\mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}_\tau , \mathcal{F}) \]
converging to the cohomology of $\mathcal{F}$ in the $\tau $-topology.
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