and define a Stokes filtration on $\mathscr{G}$ by $\mathscr{G}_{\leqslant c,\theta}:=\bigoplus_{c'\leqslant_\theta c}\mathscr{G}_{c'}$.

\[graded classification\] The set of isomorphism classes of Stokes filtered local systems $(\L,\L_\bullet)$ with an isomorphism from the graded part ${\mathrm{gr}}\L$ to $\mathscr{G}$ is identified with $H^1(S^1;\mathcal{A}ut^{<0}(\mathscr{G}))$.

The proof is standard and shown in more general contexts. Here, we only give the construction of the Stokes filtered local system from a class $\alpha \in H^1(S^1;\mathcal{A}ut^{<0}(\mathscr{G}))$. Let $(I_k)_{k=1}^\ell$ be a covering on $S^1$ by open intervals such that $I_k\cap I_{k'}=\emptyset$ if $|k-k'|\neq 1, \ell-1$. Then $\alpha$ is represented by a Čech cocycle $(\alpha_k)_{k=1}^\ell$ where $\alpha_k\in H^0(I_k\cap I_{k+1};\mathcal{A}ut^{<0}(\mathscr{G}))$ for $k=1,2,\dots,\ell-1$ and $\alpha_\ell\in H^0(I_\ell\cap I_{1};\mathcal{A}ut^{<0}(\mathscr{G}))$. Then, by gluing, there exists a unique (up to isomorphisms) local system $\L$ with isomorphisms $\beta_k: \L|_{I_k}{\xrightarrow{\sim}}\mathscr{G}|_{I_k}$ such that $\alpha_k=\beta_{k+1}(\beta_k)^{-1}$ for $k=1,2,\dots, \ell-1$ and $\alpha_\ell=\beta_1\circ (\beta_\ell)^{-1}$. The Stokes filtration on $\L$ is defined via $\beta_k$ and it is well defined and its graded part ${\mathrm{gr}}\L$ is isomorphic to $\mathscr{G}$ since $\alpha_k$ is a local section of $\mathcal{A}ut^{<0}(\mathscr{G})$.

The functor $\mathfrak{A}_{\theta_\circ}:{\sf St}^C_{\bm k}\to {\sf Std}_{\bm k}(C,\tau_{\theta_\circ})$ constructed in Proposition $\ref{object}$ is essentially surjective, and hence gives an equivalence of categories.

Let $\big{(}(V,T),(V_c,T_c)_{c\in C}, f, f^*\big{)}$ be an object in ${\sf Std}_{\bm k}(C,\tau_{\theta_\circ})$. Let $\mathscr{G}_c$ be the local system 