Lemma 21.48.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed space. The category of complexes of $\mathcal{O}$-modules with tensor product defined by $\mathcal{F}^\bullet \otimes \mathcal{G}^\bullet = \text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{G}^\bullet )$ is a symmetric monoidal category.
Proof. Omitted. Hints: as unit $\mathbf{1}$ we take the complex having $\mathcal{O}$ in degree $0$ and zero in other degrees with obvious isomorphisms $\text{Tot}(\mathbf{1} \otimes _\mathcal {O} \mathcal{G}^\bullet ) = \mathcal{G}^\bullet $ and $\text{Tot}(\mathcal{F}^\bullet \otimes _\mathcal {O} \mathbf{1}) = \mathcal{F}^\bullet $. to prove the lemma you have to check the commutativity of various diagrams, see Categories, Definitions 4.43.1 and 4.43.9. The verifications are straightforward in each case. $\square$
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