Lemma 85.17.4. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Then we have a canonical isomorphism

for $E \in D(\mathcal{C})$.

Lemma 85.17.4. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Then we have a canonical isomorphism

\[ R\Gamma (\mathcal{C}, E) = R\Gamma ((\mathcal{C}/K)_{total}, a^{-1}E) \]

for $E \in D(\mathcal{C})$.

**Proof.**
This follows from Lemma 85.17.3 because $R\Gamma ((\mathcal{C}/K)_{total}, -) = R\Gamma (\mathcal{C}, -) \circ Ra_*$ by Cohomology on Sites, Remark 21.14.4.
$\square$

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