• The choice above is such that if $M^\bullet$ has a left dual $N^\bullet$ as in Lemma 15.71.3, then we have a canonical isomorphism

$\text{Tot}(K^\bullet \otimes _ R N^\bullet ) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , K^\bullet )$

defined without the intervention of signs sending the summand $K^ p \otimes _ R N^ q$ to the summand $\mathop{\mathrm{Hom}}\nolimits _ R(M^{-q}, K^ p)$ via $N^ q = \mathop{\mathrm{Hom}}\nolimits _ R(M^{-q}, R)$ and the canonical map $K^ p \otimes _ R \mathop{\mathrm{Hom}}\nolimits _ R(M^{-q}, R) \to \mathop{\mathrm{Hom}}\nolimits _ R(M^{-q}, K^ p)$.

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