Consider a commutative diagram of commutative groups
B ⟶ C ⟶ D ↓ ↓ ↓ d A ′ ⟶ B ′ ⟶ C ′ ⟶ D ′ ↓ ↓ ↓ A ′ ′ → u ′ ′ B ′ ′ ⟶ C ′ ′ a ′ ′ ↓ A ′ ′ ′ → u ′ ′ ′ B ′ ′ ′
\begin{array}{ccccccc}
B & \longrightarrow & C & \longrightarrow & D \\
\downarrow & & \downarrow & & \downarrow d \\
A' & \longrightarrow & B' & \longrightarrow & C' & \longrightarrow & D' \\
\downarrow & & \downarrow & & \downarrow \\
A'' & \xrightarrow{u''} & B'' & \longrightarrow & C'' \\
a'' \downarrow & & & & \\
A''' & \xrightarrow{u'''} & B'''
\end{array}
B ↓ A ′ ↓ A ′′ a ′′ ↓ A ′′′ ⟶ ⟶ u ′′ u ′′′ C ↓ B ′ ↓ B ′′ B ′′′ ⟶ ⟶ ⟶ D ↓ d C ′ ↓ C ′′ ⟶ D ′ in which the rows and columns are assumed to be exact, d d d u ′ ′ u'' u ′′ a ′ ′ a'' a ′′ u ′ ′ ′ u''' u ′′′