Let $A,B_i,C$, for $i \in I$ an infinite index set, be abelian groups such that we have group homomorphisms $$r_i: A \rightarrow B_i, \,\,\, s_i: B_i \rightarrow C$$ and a complex $$A \xrightarrow{\bigoplus r_i} \bigoplus _{i \in I}B_i \xrightarrow{\sum s_i} C,$$ that is, $\mathrm{Im}(\bigoplus r_i) \subset \mathrm{Ker}(\sum s_i)$. Since $\bigoplus_{i \in I}B_i$ is an infinite direct sum, each element is a tuple $(b_i)_{i}$ such that $b_i = 0$ for all but finitely many $i$. Hence the sum $\sum s_i$ gives a well-defined element of $C$.
However, does it necessarily follow that the map $r_i:A \rightarrow B_i$ is zero for all but finitely many $i$? Is it possible that we have infinitely many nonzero $r_i$ but yet every $a \in A$ maps to an element $(b_i)_i$ such that $b_i = 0$ for all but finitely many $i$?