When we have a double complex of vector spaces $V^{p,q}$, we can produce a complex either taking direct sums or products along the anti-diagonals. Then, the differential in this new complex will be $$ d = d_{\text{vert}} + (-1)^{\text{vertical degree}} d_{\text{horizontal}}.$$ If the complex was made of complexes, I would reckon that the differential of them should appear in the differential of the double complex. However, I wasn’t able to find a reference for such a definition. Can anyone help me or point me to the definition?

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away to assign signs(:)!) fix $j$ first, write down the totalization with respect to $(p,q)$ and then do it again with $j$ added. In each step you only have to deal with a bigrading. $\endgroup$ – Sándor Kovács Jun 12 at 16:43