The Stacks project

Lemma 20.48.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. The category of complexes of $\mathcal{O}_ X$-modules with tensor product defined by $\mathcal{F}^\bullet \otimes \mathcal{G}^\bullet = \text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{G}^\bullet )$ is a symmetric monoidal category (for sign rules, see More on Algebra, Section 15.72).

Proof. Omitted. Hints: as unit $\mathbf{1}$ we take the complex having $\mathcal{O}_ X$ in degree $0$ and zero in other degrees with obvious isomorphisms $\text{Tot}(\mathbf{1} \otimes _{\mathcal{O}_ X} \mathcal{G}^\bullet ) = \mathcal{G}^\bullet $ and $\text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}_ X} \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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