The Stacks project

Example 48.3.9. Let $f : X \to Y$ be a proper morphism of Noetherian schemes, $L \in D^-_{\textit{Coh}}(X)$ and $K \in D^+_{\mathit{QCoh}}(\mathcal{O}_ Y)$. Then the map $Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, a(K)) \to R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*L, K)$ is an isomorphism. Namely, the complexes $L$ and $Rf_*L$ are pseudo-coherent by Derived Categories of Schemes, Lemmas 36.10.3 and 36.11.3 and the discussion in Remark 48.3.8 applies.