As a special case, we have the following.

\[2.16\] Set $\Delta:=(w_0)_R\in \mathrm{Br}_m$ where $w_0\in S_m$ is the longest element, i.e., $$w_0=
                        \begin{pmatrix}
                        1 & 2&\cdots & m\\
                        m& m-1&\cdots& 1
                        \end{pmatrix}.$$ Then we have $\Delta f =(f^*)^{-1}$, and $\Delta f^*=(\bigoplus_cT_c)\circ f^{-1}\circ T^{-1}$.

Set $\Delta f^!:=(\bigoplus_cT_c)^{-1}\circ \Delta f^* \circ T$. We show $\Delta f \circ f^*\circ f=f$, and $f^!\circ f\circ \Delta f^!= f^!$. We use the notation $c_i=\tau^{-1}(i)\ (i=1,\dots , m)$. Then the restriction $\Delta f\circ f^* \circ f\mid_{V_{c_i}}$ is $$\begin{aligned}
              &\sum_{i\leq j} R_m\circ \cdots \circ R_{j+1}\circ f_{c_j} \circ f^*_{c_j}\circ f_{c_i}\\
              &=R_m\circ \cdots \circ R_{i+1} \circ f_{c_i}+
                \sum_{i< j} R_m\circ \cdots \circ R_{j+1}\circ f_{c_j} \circ f^*_{c_j}\circ f_{c_i}\\
              &=R_m\circ \cdots\circ  R_{i+1} \circ f_{c_i}+\sum_{i<j\leq m}R_m\circ \cdots R_{j+1}\circ f_{c_i}
                  -\sum_{i<j\leq m}R_m\circ \cdots \circ R_j\circ f_{c_i}\\
              &=f_{c_i}.
            \end{aligned}$$ We also see that the composition $f^!_{c_i}\circ f\circ \Delta f^!$ is $$\begin{aligned}
              &\sum_{i\leq j}f^!_{c_i}\circ f_{c_j}\circ f^!_{c_j}\circ L_{j+1}\circ\cdots L_m \\
              &=f^!_{c_i}\circ L_{i+1}\circ \cdots \circ L_m+\sum_{i<j\leq m}f^!_{c_i}\circ L_{j+1}\circ \cdots \circ L_m 
                  -\sum_{i<j\leq m}f^!_{c_i}\circ L_{j}\circ \cdots \circ L_m\\
              &=f^!_{c_i}.
            \end{aligned}$$ This implies the proposition.

For a endomorphism $g\in {\mathrm{End}}(\bigoplus_c V_c)$ and $(c, c')\in C\times C$, let $g_{c c'}$ denote the component $V_c\to V_{c'}$. Using this notation, we define $$\begin{aligned}
           &{\mathrm{End}}^\diamond\left(\bigoplus_{c\in C}V_c\right):=\left\{g\in {\mathrm{End}}\left(\bigoplus_{c\in C}V_c\right)
                             \middle| g_{cc}={\mathrm{id}}_{V_c}\text{ for all } c\in C\right\}, 