The Stacks project

Remark 15.71.7. Let us explain why the sign used in the direct construction in the proof of Lemma 15.71.6 agrees with the sign we get from the construction using the discussion in Remark 15.71.2 and Categories, Remark 4.43.12. Denote $- \otimes - = \text{Tot}(- \otimes _ R -)$ and $hom(-, -) = \mathop{\mathrm{Hom}}\nolimits ^\bullet (-, -)$. The construction using monoidal category language tells us to use the arrow

in $\text{Comp}(R)$ corresponding to the arrow

gotten by swapping the order of the last two tensor products and then using the evaluation maps $hom(K^\bullet , L^\bullet ) \otimes K^\bullet \to L^\bullet $ and $hom(L^\bullet , K^\bullet ) \otimes L^\bullet \to M^\bullet $. Only in swapping does a sign intervene. Namely, in the isomorphism

there is a sign $(-1)^{r(q + r')}$ on $K^ r \otimes _ R \mathop{\mathrm{Hom}}\nolimits _ R(K^{-r'}, L^ q)$, see Section 15.72 item (9). The reader can convince themselves that, because of the correspondence we are using to describe maps into an internal hom, this sign only matters if $r = r'$ and in this case we obtain $(-1)^{r(q + r)} = (-1)^{r + qr}$ as in the direct proof.