The Stacks project

Remark 46.10.3. Remark 46.10.2 above implies we have an equivalence of derived categories

where $\mathcal{C}$ is the category of parasitic adequate modules. Namely, it is clear that $D_\mathcal {C}(\mathcal{O})$ is the kernel of $Rf_*$, hence a functor as indicated. For any object $X$ of $D_{\textit{Adeq}}(\mathcal{O})$ the map $Lf^*Rf_*X \to X$ maps to a quasi-isomorphism in $D_\mathit{QCoh}(S)$, hence $Lf^*Rf_*X \to X$ is an isomorphism in $D_{\textit{Adeq}}(\mathcal{O})/D_\mathcal {C}(\mathcal{O})$. Finally, for $X, Y$ objects of $D_{\textit{Adeq}}(\mathcal{O})$ the map

is bijective as $Lf^*$ gives an inverse (by the remarks above).