The Stacks project

Lemma 21.43.5. In the situation above, suppose that $K$ is an object of $\mathit{QC}(\mathcal{O})$ and $M$ arbitrary in $D(\mathcal{O})$. For every object $U$ of $\mathcal{C}$ we have

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(K|_ U, M|_ U) = R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}(U)}(R\Gamma (U, K), R\Gamma (U, M)) \]

Proof. We may replace $\mathcal{C}$ by $\mathcal{C}/U$. Thus we may assume $U = X$ is a final object of $\mathcal{C}$. By Lemma 21.43.4 we see that $K = Lf^*P$ where $P = R\Gamma (U, K) = R\Gamma (X, K) = Rf_*K$. Thus the result because $Lf^*$ is the left adjoint to $Rf_*(-) = R\Gamma (U, -)$. $\square$

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