Proposition 5.16, A sequence $$\begin{equation*}M'\xrightarrow{\alpha}M\xrightarrow{\beta}M''\end{equation*}\tag{1}$$ is exact if and only if $$\begin{equation*}M_{\mathfrak{m}}'\xrightarrow{\alpha_{\mathfrak{m}}}M_{\mathfrak{m}}\xrightarrow{\beta_{\mathfrak{m}}} M_{\mathfrak{m}}''\end{equation*}\tag{2}$$ is exact for all maximal ideals $\mathfrak{m}$.
Proof given in this book: only if statement is clear recalling the fact that $M\mapsto M_{\mathfrak{m}}$ is exact functor. To show the converse, let $N = \text{Ker}(\beta)/\text{Im}(\alpha)$. Because the functor $M\mapsto M_{\mathfrak{m}}$ is exact, $$N_{\mathfrak{m}}=\text{Ker}(\beta_{\mathfrak{m}})/\text{Im}(\alpha_{\mathfrak{m}}).$$ If $(2)$ is exact for all $\mathfrak{m}$, then $N_{\mathfrak{m}}=0$ for all $\mathfrak{m}$, then $N=0$ (by the local property). Hence $(1)$ is exact.
I don't understand the statement '$M\mapsto M_{\mathfrak{m}}$ is exact, $N_{\mathfrak{m}}=\text{Ker}(\beta_{\mathfrak{m}})/\text{Im}(\alpha_{\mathfrak{m}})$.' Could you explain this?