Remark 20.38.12. Let $h : X \to Y$ be a morphism of ringed spaces. Let $K, L$ be objects of $D(\mathcal{O}_ Y)$. We claim there is a canonical map

$Lh^*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (Lh^*K, Lh^*L)$

in $D(\mathcal{O}_ X)$. Namely, by (20.38.0.1) proved in Lemma 20.38.2 such a map is the same thing as a map

$Lh^*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \otimes ^\mathbf {L} Lh^*K \longrightarrow Lh^*L$

The source of this arrow is $Lh^*(\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \otimes ^\mathbf {L} K)$ by Lemma 20.27.3 hence it suffices to construct a canonical map

$R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \otimes ^\mathbf {L} K \longrightarrow L.$

For this we take the arrow corresponding to

$\text{id} : R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$

via (20.38.0.1).

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