Lemma 21.35.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\mathcal{O})$. The construction of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ commutes with restrictions, i.e., for every object $U$ of $\mathcal{C}$ 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 21.20.1.
$\square$

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