an isomorphism since it is deduced from the condition $(1)$ and the condition that $f$ is an isomorphism. Similarly, we can show that $f^!:= \prod_cf_c^! :V\to\bigoplus_c V_c$ is an isomorphism.

Let ${\sf Std}_{\bm k}({C},\tau)$ denote the category of representations in ${{\sf Rep}}_{\mathbb{Z}}({\bm k})$ with Stokes data of type $({C},\tau)$ defined above (a morphism in this category is defined as a morphism in ${{\sf Rep}}_{\mathbb{Z}}({\bm k})$ compatible with the Stokes data). Fix a $m$-point set $C\subset {\mathbb{C}}$ and $C$-generic point $\theta_\circ \in {\mathbb{R}}$. Then we have a unique bijection $\tau_{\theta_\circ}:C{\xrightarrow{\sim}}\{1,2,\dots, m\}$ with the following property: when we put $c_i:=\tau^{-1}_{\theta_\circ}(i)$ for $i=1,2,\dots, m$, we have $c_1<_{\theta_{\circ}+\pi/2}c_2<_{\theta_{\circ}+\pi/2}\cdots <_{\theta_{\circ}+\pi/2}c_m$. In the rest of §\[stofil\], we shall construct the functor $\mathfrak{A}_{\theta_\circ}:{\sf St}_{\bm k}^C\to{\sf Std}_{\bm k}(C,\tau_{\theta_\circ})$ and show that $\mathfrak{A}_{\theta_\circ}$ gives an equivalence of categories.

To prepare for the construction of the functor, we recall the equivalence ${\sf Loc}_{\bm k}{\xrightarrow{\sim}}{\sf Rep}_{\bm k}$ of categories. Let $p:{\mathbb{R}}\to S^1:={\mathbb{R}}/2\pi{\mathbb{Z}}$ be the quotient map. This gives an universal covering of $S^1$. For $\L\in {\sf Loc}_{\bm k}$, set $V_\L:=\Gamma({\mathbb{R}}, p^{-1}\L)$ and define the map $T_\L:V_\L\to V_\L$ by $T_\L v(\theta):=v(\theta+2\pi)$. The correspondence $\L\mapsto (V_\L,T_\L)$ is functorial and we can construct the quasi-inverse functor explicitly, this is an equivalence of categories.

By 