There is a canonical isomorphism
\[ \text{Tot}(M^\bullet \otimes _ R N^\bullet )[a + b] \to \text{Tot}(M^\bullet [a] \otimes _ R N^\bullet [b]) \]which uses the sign $(-1)^{pb}$ on the summand $M^ p \otimes _ R N^ q$, see Homology, Remark 12.18.5. It is often more convenient to consider the corresponding shifted map $\text{Tot}(M^\bullet \otimes _ R N^\bullet ) \to \text{Tot}(M^\bullet [a] \otimes _ R N^\bullet [b])[-a - b]$.
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