Remark 12.18.5. Let $\mathcal{A}$ be an additive category. Let $A^{\bullet , \bullet }$ be a double complex with differentials $d_1^{p, q}$ and $d_2^{p, q}$. Denote $A^{\bullet , \bullet }[a, b]$ the double complex with

and differentials

In this situation there is a well defined isomorphism

which in degree $n$ is given by the map

for some sign $\epsilon (p, q, a, b)$. Of course the summand $A^{p, q}$ maps to the summand $A^{p' + a, q' + b}$ when $p = p' + a$ and $q = q' + b$. To figure out the conditions on these signs observe that on the source we have

whereas on the target we have

Thus our constraints are that

and

Thus we choose $\epsilon (p, q, a, b) = (-1)^{pb}$.

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