Suppose we have a homomorphism $\alpha, \beta, \gamma$ of short exact sequences: $$ \begin{matrix} 0 & \to & A & \xrightarrow{\psi} & B & \xrightarrow{\phi} & C & \to & 0 \\ \ & \ & \downarrow^{\alpha} & \ & \downarrow^{\beta} \ & \ & \downarrow^{\gamma} \\ 0 & \to & A' & \xrightarrow{\psi'} & B' & \xrightarrow{\phi'} & C' & \to & 0 \end{matrix} $$
If both $\alpha, \gamma$ are surjective then so is $\beta$. This can be proved using the properties of the diagram somehow.
I've tried several things.