corresponding to $(V_c,T_c)$. Set $\mathscr{G}:=\bigoplus_{c\in C}\mathscr{G}_c$. Define a local endomorphism $\lambda^*:\mathscr{G}|_{p(I_{\theta_\circ-\pi}\cap I_{\theta_\circ})}
          \to \mathscr{G}|_{p(I_{\theta_\circ-\pi}\cap I_{\theta_\circ})}$ by the following composition: $$\Gamma(I_{\theta_\circ-\pi}\cap I_{\theta_\circ}, p^{-1}\mathscr{G}) 
          \xleftarrow{\sim}  \bigoplus_c V_c 
          \xrightarrow{f^*\circ f} \bigoplus_c V_c
          {\xrightarrow{\sim}}\Gamma(I_{\theta_\circ-\pi}\cap I_{\theta_\circ}, p^{-1}\mathscr{G})$$ where the first and the third isomorphisms are the restriction maps regarding $V_c$ as $\Gamma({\mathbb{R}}, p^{-1}\mathscr{G}_c)$. By Definition \[stokes data\] (1), $\lambda^*$ is a local section of $\mathcal{A}ut^{<0}(\mathscr{G})$ on $p(I_{\theta_\circ-\pi}\cap I_{\theta})$. Similarly, we can define a local endomorphism $\lambda^!:\mathscr{G}|_{p(I_{\theta_\circ-\pi}\cap I_{\theta})}
          \to \mathscr{G}|_{p(I_{\theta_\circ-\pi}\cap I_{\theta})}$ by using $f^!\circ f$ and it defines a local section of $\mathcal{A}ut^{<0}(\mathscr{G})$ on $p(I_{\theta_\circ-\pi}\cap I_{\theta})$. The pair $(\lambda^*,\lambda^!)$ defines a class in $H^1(S^1;\mathcal{A}ut^{<0}(\mathscr{G}))$ as Čech cocycle for the covering $S^1=p(I_{\theta_\circ})\cup p(I_{\theta_{\circ}+\pi})$. By Lemma \[graded classification\], it defines a Stokes filtered local system $(\L,\L_\bullet)$. The fact that $\mathfrak{A}_{\theta_\circ}(\L,\L_\bullet)\simeq \big{(}(V,T),(V_c,T_c)_{c\in C}, f, f^*\big{)}$ is obvious by the construction of $\mathfrak{A}_{\theta_\circ}$ and the proof of Lemma \[graded classification\].

By this theorem, the notion of Stokes data introduced here is equivalent to the notion of Stokes data introduced in [@herexa].

Mutations on Stokes data {#mutations on std}
------------------------

Let $\big{(}(V,T),(V_{c},T_{c})_{{c}\in {C}}, f, f^*\big{)}$ be an object in ${\sf Std}_{\bm k}({C},\tau)$. For each $i\in\{1,2,\dots, m\}$, we define the endomorphisms $R_i,R_i^*,L_i,L_i^!$ on $V$ as follows: $$\begin{aligned}
                R_i&:={\mathrm{id}}_V-f_{\tau^{-1}(i)}\circ f_{\tau^{-1}(i)}^*, \\
                R_i^*&:={\mathrm{id}}_V-T\circ f_{\tau^{-1}(i)}\circ 