The Stacks project

Example 86.3.6. Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of Noetherian algebraic spaces over $S$, $L \in D^-_{\textit{Coh}}(X)$ 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 Spaces, Lemmas 75.13.7 and 75.8.1 and the discussion in Remark 86.3.5 applies.