Let $f:(A,d) \rightarrow (A^{'},d^{'})$ be a chain map. For each $n$ define $$M_{n}=A_{n-1} \oplus A^{'}_n$$ and $\Delta_{n} :M_{n} \rightarrow M_{n-1}$ by $$\Delta_{n}:(a_{n-1},a_{n}^{'}) \rightarrow (-d_{n-1}a_{n-1},d^{'}a^{'}+f_{n-1}a_{n-1}).$$
Prove that $(M,\Delta)$ as just defined is a complex.
this will be great if you give me some hint or Idea,thanks a lot.