## 21.35 Global derived hom

Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $K, L \in D(\mathcal{O})$. Using the construction of the internal hom in the derived category we obtain a well defined object

in $D(\Gamma (\mathcal{C}, \mathcal{O}))$. By Lemma 21.34.1 we have

and

If $f : (\mathcal{C}', \mathcal{O}') \to (\mathcal{C}, \mathcal{O})$ is a morphism of ringed topoi, then there is a canonical map

in $D(\Gamma (\mathcal{O}))$ by taking global sections of the map defined in Remark 21.34.11.

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