Lemma 38.3 (Barrett and Whitehead):

Given a commutative diagram with exact rows,

$$\begin{CD} @> >> A_n @> {p_n} >> B_n @> q_n >> C_n @> r_n >> A_{n... ...n^{\prime}>> C_n^{\prime} @> r_n^{\prime}>> A_{n-1}^{\prime} @> >> \\ \end{CD}$$

in If each of the maps $\gamma_n:C_n\longrightarrow C_n^{\prime}$ is an isomorphism, then the sequence

$$\begin{CD} @> >> A_n @> \lambda_n >> B_n\oplus A_n^{\prime} @> \mu_n >> B_n^{\prime} @> \delta_n >> A_{n-1}@> >> \end{CD}$$

is exact where, the maps are given by

$$\lambda_n = (p_n, -\alpha_n),\quad \mu_n = \beta_n + p_n^{\prime}, \quad \delta_n = r_n\circ \gamma_n^{-1}\circ q_n^{\prime}.$$