The Stacks project

Proof. We will prove a right adjoint exists by verifying the hypotheses of Derived Categories, Proposition 13.38.2. First off, the category $D_\mathit{QCoh}(\mathcal{O}_ X)$ has direct sums, see Derived Categories of Spaces, Lemma 75.5.3. The category $D_\mathit{QCoh}(\mathcal{O}_ X)$ is compactly generated by Derived Categories of Spaces, Theorem 75.15.4. Since $X$ and $Y$ are quasi-compact and quasi-separated, so is $f$, see Morphisms of Spaces, Lemmas 67.4.10 and 67.8.9. Hence the functor $Rf_*$ commutes with direct sums, see Derived Categories of Spaces, Lemma 75.6.2. This finishes the proof. $\square$