Proposition \[split\], (1), we have a unique splitting $\eta_{[I_{\theta_\circ}]}:\L|_{[I_{\theta_\circ}]}{\xrightarrow{\sim}}{\mathrm{gr}}\L|_{[I_{\theta_\circ}]}$. We put $\eta_{\theta_\circ}:=(p^{-1}\eta_{[I_{\theta_\circ}]})|_{I_{\theta_\circ}}:(p^{-1}\L)|_{I_{\theta_\circ}}
          {\xrightarrow{\sim}}(p^{-1}{\mathrm{gr}}\L)|_{I_{\theta_\circ}}$, which is an isomorphism of sheaves on $I_{\theta_\circ}\subset {\mathbb{R}}$.

For an object $(\L,\L_\bullet)\in {\sf St}_{\bm k}^C$, we have $(V_\L,T_\L)$ and $(V_{c},T_c):=(V_{{\mathrm{gr}}_{c}\L},T_{{\mathrm{gr}}_c\L})$ for $c\in C$. Let $r_{{\mathrm{gr}}\L}^{\theta}:\bigoplus_cV_c{\xrightarrow{\sim}}\Gamma(I_{\theta}, p^{-1}{\mathrm{gr}}\L)$ and $r^\theta_\L:V_\L{\xrightarrow{\sim}}\Gamma(I_{\theta},p^{-1}\L)$ be the restriction maps for $\theta\in{\mathbb{R}}$. We define $f_{\theta_\circ}:\bigoplus_{c\in C}V_c{\xrightarrow{\sim}}V$ as a composition: $f_{\theta_\circ}:=
          (r_\L^{\theta_\circ})^{-1}
          \circ\Gamma(I_{\theta_\circ},\eta_{\theta_\circ}^{-1})
          \circ r^{\theta_\circ}_{{\mathrm{gr}}\L}$. Similarly, we define $f^*_{\theta_\circ}:V{\xrightarrow{\sim}}\bigoplus_{c\in C}V_c$ as a composition: $f^*_{\theta_\circ}:=
          (r_{{\mathrm{gr}}\L}^{\theta_\circ-\pi})^{-1}
          \circ \Gamma(I_{\theta_\circ-\pi},\eta_{\theta_\circ-\pi})
          \circ r^{\theta_\circ-\pi}_\L$. Then, $f^!_{\theta_\circ}:=\bigoplus_c T^{-1}_c\circ f^*_{\theta_\circ}\circ T$ is described as a composition: $f^!_{\theta_\circ}:=
          (r_{{\mathrm{gr}}\L}^{\theta_\circ+\pi})^{-1}
          \circ \Gamma(I_{\theta_\circ+\pi},\eta_{\theta_\circ+\pi})
          \circ r^{\theta_\circ+\pi}_\L$.

\[object\] The data $\big{(}(V_c,T_c)_{c\in C},f_{\theta_\circ},f^*_{\theta_\circ}\big{)}$ are Stokes data on $(V_\L,T_\L)$ of type $(C,\tau_{\theta_\circ})$. Moreover, the correspondence $(\L,\L_\bullet)\mapsto \big{(}(V_\L,T_\L),(V_c,T_c)_{c\in C},f_{\theta_\circ},f^*_{\theta_\circ}\big{)}$, and $\lambda
           \mapsto \Gamma({\mathbb{R}}, p^{-1}\lambda)$ $($where $\lambda$ is a morphism in ${\sf St}^C_{\bm k}$$)$ gives a fully faithfull functor $\mathfrak{A}_{\theta_\circ}:
           {\sf St}^C_{\bm k}\to {\sf Std}_{\bm k}(C,\tau_{\theta_\circ})$.

Let $r^{\theta,\theta'}: V_{{\mathrm{gr}}\L}\to \Gamma (I_\theta\cap I_{\theta'},{\mathrm{gr}}\L)$ be the restriction map for $\theta,\theta'\in {\mathbb{R}}$. The composition $f^*_{\theta_\circ}\circ f_{\theta_\circ}$ can be described as the composition: $$(r^{\theta_\circ,\theta_\circ-\pi})^{-1} \circ
          \Gamma(I_{\theta_\circ}\cap I_{\theta_\circ-\pi},\eta_{\theta_\circ-\pi})\circ
          \Gamma(I_{\theta_\circ}\cap I_{\theta_\circ-\pi},\eta_{\theta_\circ}^{-1})\circ
           r^{\theta_\circ,\theta_\circ-\pi}.$$ The fact that $\eta_{\theta_\circ-\pi}$ and $\eta_{\theta_\circ}$ are splitting implies $f^*_{\theta_\circ, c}\circ f_{\theta_\circ, c}={\mathrm{id}}_{V_c}$. Since $\eta_{\theta_\circ-\pi}$ and $\eta_{\theta_\circ}$ are compatible with filtration, we have $f^*_{\theta_\circ, c'}\circ f_{\theta_\circ, c}=0$ if $c <_{\theta_\circ-\pi/2} c'$. This proves the condition 