Lemma 20.42.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L$ be objects of $D(\mathcal{O}_ X)$. The construction of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ commutes with restrictions to opens, i.e., for every open $U$ we have $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K|_ U, L|_ U) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)|_ U$.

**Proof.**
This is clear from the construction and Lemma 20.32.1.
$\square$

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