This is related to Weibel Exercise 2.7.1 $C$ is periodic upper half plan complex $C_{pq}=Z_4$ for all $p\in Z,q\geq 0$ all differentials are multiplication by 2.
It is easy to apply acyclic assembly lemma to see $H_i(Tot^{\oplus}(C))=0$ for all $i$. It is not too hard to compute $H_0(Tot^{\prod}(C))=Z_2$.
$\textbf{Q:}$ Since vertical direction is not exact and I cannot cut off by truncation at $q=0$ to replace by kernel, I cannot use acyclic assembly lemma to conclude vanishing of $H_i(Tot^{\prod}(C))$ for $i\neq 0$. If I draw the line $p+q=n$, it will always hit the bottom $q=0$ part of the complex, so truncation does not make too much sense here. Is $H_i(Tot^{\prod}(C))=0$ for $i\neq 0$ or was there trick to compute them?