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 Schemes, Lemma 36.3.1. The category $D_\mathit{QCoh}(\mathcal{O}_ X)$ is compactly generated by Derived Categories of Schemes, Theorem 36.15.3. Since $X$ and $Y$ are quasi-compact and quasi-separated, so is $f$, see Schemes, Lemmas 26.21.13 and 26.21.14. Hence the functor $Rf_*$ commutes with direct sums, see Derived Categories of Schemes, Lemma 36.4.5. This finishes the proof. $\square$