The Stacks project

Lemma 15.58.1. Let $R$ be a ring. The category $\text{Comp}(R)$ of complexes of $R$-modules endowed with the functor $(L^\bullet , M^\bullet ) \mapsto \text{Tot}(L^\bullet \otimes _ R M^\bullet )$ and associativity and commutativity constraints as above is a symmetric monoidal category.

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