Lemma 17.15.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_ Y$-modules. Then $f^*(\mathcal{F} \otimes _{\mathcal{O}_ Y} \mathcal{G}) = f^*\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G}$ functorially in $\mathcal{F}$, $\mathcal{G}$.

**Proof.**
Omitted.
$\square$

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