The Stacks project

Lemma 21.43.4. In the situation above, suppose that $\mathcal{C}$ has a final object $X$. Set $R = \mathcal{O}(X)$ and denote $f : (\mathcal{C}, \mathcal{O}) \to (pt, R)$ the obvious morphism of sites. Then $\mathit{QC}(\mathcal{O}) = D(R)$ given by $Lf^*$ and $Rf_*$.

Proof. Omitted. $\square$

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