Direct sum and product totalisations are always isomorphic

All double-complex identities hold because all maps vanish. The degree-zero direct-sum total object is $S$ and the degree-zero product total object is $P$ by [F1, F2]. Every other total degree is zero and both total differentials are zero. Thus both totalisations exist, with infinitely many nonzero summands on their sole nonzero diagonal.

An isomorphism of these chain complexes would induce a bijection $S\to P$, impossible because $S$ is countably infinite and $P$ is uncountable. The canonical comparison is also explicitly nonsurjective: the constant-one sequence lies in $P$ and has infinite support, so is absent from $S$. This verifies the failed conclusion for abstract as well as canonical isomorphisms. All zero degrees and the index $j=0$ are included; the cardinality proof and this witness require no AC.