any $y\in Y$, $$\begin{aligned}
([\psi_{(1)}\rightharpoonup T][\psi_{(2)}\rightharpoonup R])(y))
& =[\psi_{(2)}\rightharpoonup R]([\psi_{(1)}\rightharpoonup T](y)) \\
& =\psi_{(3)}\triangleright R(S^0 (\psi_{(2)})\triangleright[\psi_{(1)}\rightharpoonup T](y)) \\
& =\psi_{(4)}\triangleright R(S^0 (\psi_{(3)})\psi_{(2)}\triangleright T(S^0 (\psi_{(1)})\triangleright y)) \\
& =\psi_{(2)}\triangleright R(T(S^0 (\psi_{(1)})\triangleright y)) \\
& =\psi_{(2)}\triangleright(TR)(S^0 (\psi_{(1)})\triangleright y) \\
& =[\psi\rightharpoonup TR](y) .\end{aligned}$$ Hence we obtain that $\psi\rightharpoonup TR=[\psi_{(1)}\rightharpoonup T][\psi_{(2)}\rightharpoonup R]$. For any $y\in Y$, $$\begin{aligned}
[\psi\rightharpoonup[\chi\rightharpoonup T]](y) & =
\psi_{(2)}\triangleright[\chi\rightharpoonup T](S^0 (\psi_{(1)})\triangleright y) \\
& =\psi_{(2)}\triangleright\chi_{(2)}\triangleright T(S^0 (\chi_{(1)})\triangleright S^0 (\psi_{(1)})\triangleright y) \\
& =\psi_{(2)}\chi_{(2)}\triangleright T(S^0 (\psi_{(1)}\chi_{(1)})\triangleright y) \\
& =[\psi\chi\rightharpoonup T](y) .\end{aligned}$$ Hence we obtain that $\psi\rightharpoonup[\chi\rightharpoonup T]=\psi\chi\rightharpoonup T$. Let $I_Y$ be the identity map on $Y$. Then for any $y\in Y$ $$[\psi\rightharpoonup I_Y ](y)=\psi_{(2)}\triangleright I_Y (S^0 (\psi_{(1)})\triangleright y)
=\epsilon^0 (\psi)y .$$ Hence $\psi\rightharpoonup I_Y =\epsilon^0 (\psi)I_Y$. Also, for any $y\in Y$, $[1^0 \rightharpoonup T](y)=T(y)$. Thus $1^0 \rightharpoonup T=T$. Furthermore, for any $y, z\in Y$, $$\begin{aligned}
{}_C {\langle}[\psi\rightharpoonup T]^* (y) \, , \, z {\rangle}& ={}_C {\langle}y \, , \, [\psi\rightharpoonup T](z) {\rangle}\\
& ={}_C {\langle}y \, , \, \psi_{(2)}\triangleright T(S^0 (\psi_{(1)})\triangleright z) {\rangle}\\
& ={}_C {\langle}S^0 (\psi_{(3)}^* )\psi_{(4)}^* \triangleright y \, , \, \psi_{(2)}
\triangleright T(S^0 (\psi_{(1)})\triangleright z) {\rangle}\\
& = S^0 (\psi_{(2)}^* )\cdot_{\widehat{\rho}}\,{}_C {\langle}\psi_{(3)}^* \triangleright y \, , \, 
T(S^0 (\psi_{(1)})\triangleright z) {\rangle}\\
& = S^0 (\psi_{(2)}^* )\cdot_{\widehat{\rho}}\,{}_C {\langle}T^* 