T_{\tau^{-1}(i)}^{-1}\circ f_{\tau^{-1}(i)}^*, \\
                L_i&:={\mathrm{id}}_V-f_{\tau^{-1}(i)}\circ f_{\tau^{-1}(i)}^!, \\
                L_i^!&:={\mathrm{id}}_V-T^{-1} \circ f_{\tau^{-1}(i)}\circ T_{\tau^{-1}(i)} \circ f_{\tau^{-1}(i)}^!.
               \end{aligned}$$

We remark that $T^{-1}\circ R_i^*\circ T=L_i$ and $T\circ L_i^!\circ T^{-1}=R_i$. We also have $R^*_i\circ R_i=R_i\circ R_i=R_i$, and $L^!_i\circ L_i=L_i\circ L_i =L_i$ by easy computation.

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 $i\in \{2,\dots, m\}$, we define $$\begin{aligned}
               ({\mathbb{R}}_i f)_{c}&:=\begin{cases}
                                R_{i}\circ f_{c}& (\tau({c})=i-1) \\
                                f_{c}& \text{otherwise}
                               \end{cases}, \\
                ({\mathbb{R}}_i f^*)_{c}&:=\begin{cases}
                                        f_{c}^*\circ R^*_i  & (\tau({c})=i-1)\\
                                       f_{c}^* & \text{otherwise}.
                                      \end{cases}
              \end{aligned}$$ Then the tuple $\big{(}(V,T),(V_{c},T_{c})_{{c}\in {C}}, {\mathbb{R}}_i f, {\mathbb{R}}_i f^*\big{)}$ is an object of ${\sf Std}_{\bm k}({C}, (i,\ i-1)\circ \tau)$. Similarly, for $i\in\{1,\dots, m-1\}$, we define $$\begin{aligned}
               (\mathbb{L}_i f)_{c}&:=\begin{cases}
                                L_{i}\circ f_{c}& (\tau ({c})=i+1) \\
                                f_{c}& \text{otherwise}
                               \end{cases}, \\
                (\mathbb{L}_i f^*)_{c}&:=\begin{cases}
                                        f_{c}^*\circ T\circ L_i^{!}\circ T^{-1}  & (\tau({c})=i+1)\\
                                       f_{c}^* & \text{otherwise}.
                                      \end{cases}
              \end{aligned}$$ Then the tuple $\big{(}(V,T),(V_{c},T_{c})_{{c}\in {C}}, \mathbb{L}_i f, \mathbb{L}_i f^*\big{)}$ is an object of ${{\sf Std}}_{\bm k}({C}, (i,\ i+1)\circ \tau)$.

We show the first half of the proposition. The second half is shown similarly. Let ${c}, {{c'}}$ be two distinct elements of $C$. We only consider the case where $\tau({c})=i-1$ and $\tau({{c'}})=i$ since the discussion is easier in the other cases.

We first show that $({\mathbb{R}}_i f)_c=R_i\circ f_c$ is injective. Let $u$ be any 