Direct sum total complex of a double complex

Let $C$ be a homological double complex in an abelian category. Suppose that for every integer $n$ the diagonal coproduct below exists, and write its injections as ${\iota }_{p,q}^n$: $T_n={\mathrm{Tot}}_n^{\oplus }(C)={\coprod }_{p+q=n}{C}_{p,q}\mathrm{.}$ Define $d_n:T_n\to T_{n-1}$ to be the unique arrow with $d_n{\iota }_{p,q}^n={\iota }_{p-1,q}^{n-1}{h}_{p,q}+{\iota }_{p,q-1}^{n-1}{v}_{p,q}(p+q=n)\mathrm{.}$ Each right-hand side is the sum of two morphisms with the same source and target. The coproduct universal property supplies a unique $d_n$ from this family; no infinite sum in a morphism group is required.

The resulting direct-sum total complex has differential of degree $-1$. Its chain condition is established in The total differential squares to zero ↗. A diagonal with only zero objects has zero coproduct: every family of maps from its objects is the unique zero family. A diagonal with exactly one nonzero object has that object as coproduct. For general infinite diagonals existence is a hypothesis, not an implication of being an abelian category. No choices of elements or representatives enter this construction.