Lemma 20.27.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. There is a canonical bifunctorial isomorphism

for $\mathcal{F}^\bullet $ in $D(\mathcal{O}_ X)$ and $\mathcal{G}^\bullet $ in $D(\mathcal{O}_ Y)$.

Lemma 20.27.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. There is a canonical bifunctorial isomorphism

\[ \mathcal{F}^\bullet \otimes _{\mathcal{O}_ X}^{\mathbf{L}} Lf^*\mathcal{G}^\bullet = \mathcal{F}^\bullet \otimes _{f^{-1}\mathcal{O}_ Y}^{\mathbf{L}} f^{-1}\mathcal{G}^\bullet \]

for $\mathcal{F}^\bullet $ in $D(\mathcal{O}_ X)$ and $\mathcal{G}^\bullet $ in $D(\mathcal{O}_ Y)$.

**Proof.**
Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module and let $\mathcal{G}$ be an $\mathcal{O}_ Y$-module. Then $\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G} = \mathcal{F} \otimes _{f^{-1}\mathcal{O}_ Y} f^{-1}\mathcal{G}$ because $f^*\mathcal{G} = \mathcal{O}_ X \otimes _{f^{-1}\mathcal{O}_ Y} f^{-1}\mathcal{G}$. The lemma follows from this and the definitions.
$\square$

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