Theorem 38.1 (The five lemma):

Assume given a commutative diagram of abelian groups with exact rows:

$\begin{CD} A_1 @> {\alpha_1} >> A_2 @> \alpha_2 >> A_3 @> \alpha_3 >> A_4 @> ... ...eta_1 >> B_2 @> {\beta_2}>> B_3 @> \beta_3>> B_4 @> \beta_4 >> B_5 \\ \end{CD}$

If $\psi_1, \psi_2, \phi_1$ and $\phi_2$ are isomorphisms then so is $\eta$.