Lemma 21.36.5. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the corresponding morphism of topoi. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

For $M$ in $D(\mathcal{D})$ we have $R\Gamma (U, g^{-1}M) = R\Gamma (u(U), M)$.

If $\mathcal{O}_\mathcal {D}$ is a sheaf of rings and $\mathcal{O}_\mathcal {C} = g^{-1}\mathcal{O}_\mathcal {D}$, then for $M$ in $D(\mathcal{O}_\mathcal {D})$ we have $R\Gamma (U, g^*M) = R\Gamma (u(U), M)$.

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