When one defines $\text{Tot}(C) = \text{Tot}^{П}(C)$ a total complex one usually writes "the formula $d=d^h+d^v$ defines map $d$: $\text{Tot}^{П}(C)_n \rightarrow \text{Tot}^{П}(C)_{n-1}$". How one gets this map $d$? (What does $d=d^h+d^v$ mean exactly?)
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2You are working with a bigraded complex $C^{\bullet,\bullet}$ with the "horizontal" $d^h$ and "vertical" $d^v$ when you lay it out in 2D. – user10354138 Apr 21 '21 at 11:31