Exercise 8, § 2

Let $0 \to M' \xrightarrow{u} M \xrightarrow{v} M'' \to 0$ be an exact sequence of A-modules, which can be considered as an exact sequence of complexes (zero in degrees $\neq 0$). Now prove that the morphism of complexes $\varphi : \mathrm{Con}\,(u) \to M''$ (X, p. 39) is a homotopy equivalence if and only if the exact sequence $0 \to M' \to M \to M'' \to 0$ is split.