In Beauville, Narasimhan, Ramanan's *Spectral curves and the generalized theta divisor*, Remark 3.7, the following exact sequence is presented:

$0 \rightarrow M(-\Delta) \rightarrow \pi^* E \xrightarrow{\pi^*\varphi - x} \pi^*(L \otimes E) \rightarrow \pi^*L \otimes M \rightarrow 0$

where $X$ is a smooth curve, $L$ is a line bundle on $X$, $(E, \varphi)$ is a $L$-twisted Higgs bundle on $X$, $\pi: X_s \rightarrow X$ is the spectral curve associated to the characteristic polynomial of $\varphi$ (supposed to be integral), $x$ is the tautological section of $\pi^*L$, $\Delta = \pi^*L^{\deg\pi-1}$ is the ramification divisor of $\pi$, and $M$ is the unique torsion free sheaf of rank 1 on $X_s$ such that $E=\pi_* M$.

Do you have any suggestion about how to prove the exactness of the sequence? The intuition is that $M(-\Delta)$ must be the eigenspace of $\pi^*\varphi$ associated to the eigenvalue $x$, but it is not clear to me why it should be.