$(1)$ of the Definition \[stokes data\] because $c <_{\theta_\circ-\pi/2} c'$ is equivalent to $\tau_{\theta_\circ}(c')<\tau_{\theta_\circ}(c)$. Similarly, the composition $f^!_{\theta_\circ}\circ f_{\theta_\circ}$ is described as the following composition: $$(r^{\theta_\circ,\theta_\circ+\pi})^{-1} \circ
          \Gamma(I_{\theta_\circ}\cap I_{\theta_\circ+\pi},\eta_{\theta_\circ+\pi})\circ
          \Gamma(I_{\theta_\circ}\cap I_{\theta_\circ+\pi},\eta_{\theta_\circ}^{-1})\circ
           r^{\theta_\circ,\theta_\circ+\pi}.$$ Since $c<_{\theta_\circ+\pi/2} c'$ is equivalent to $\tau_{\theta_\circ}(c)<\tau_{\theta_\circ}(c')$, we have the condition $(2)$ of Definition \[stokes data\].

Let $(\L,\L_\bullet)$ and $(\L',\L'_\bullet)$ be objects in ${\sf St}^C_{\bm k}$. Let $\lambda:\L\to\L'$ be a morphism of local systems. It induces a morphism $g:=\Gamma({\mathbb{R}}, p^{-1}\lambda):(V_\L,T_\L)\to (V_{\L'},T_{\L'})$. Proposition \[split\], (2) implies that $\lambda$ is compatible with Stokes filtration if and only if $g$ is compatible with Stokes data constructed above. This shows that the correspondence gives a fully faithfull functor $\mathfrak{A}_{\theta_\circ}:{\sf St}^C_{\bm k}\to {\sf Std}_{\bm k}(C,\tau_{\theta_\circ})$.

We shall prove the the essential surjectivity of $\mathfrak{A}_{\theta_\circ}$. We recall a classification result of Stokes filtered local systems with a fixed graded Stokes structure. For a Stokes filtered local system $(\L,\L_\bullet)$, $\mathscr{E}nd(\L)$ is equipped with a natural Stokes filtration. We put $\mathcal{A}ut^{<0}(\L):={\mathrm{id}}_\L+\mathscr{E}nd(\L)_{<0}\subset \mathscr{E}nd(\L)_{\leqslant0}$. A local section of $\mathcal{A}ut^{<0}(\L)$ is a local endomorphism $\lambda:\L\to \L$ of the local system which is compatible with the Stokes filtration such that the graded morphism ${\mathrm{gr}}(\lambda)$ is identity. For a set of local systems $(\mathscr{G}_c)_{c\in C}$ indexed by $C$, put $\mathscr{G}:=\bigoplus_{c\in C}\mathscr{G}_c$ 