Orthogonality relations for Haar state and antipode (Timmerman) Consider the following proposition from Timmerman's "An invitation to quantum groups and duality":

I am having trouble seeing why the boxed equations are true (Note that on the left the symbol $S$ denotes the antipode, and on the right it denotes the mentioned linear map, so perhaps another notation for $S$ and $T$ would have been less confusing).
In particular, where did the antipode on the right hand side go to?
 A: The short answer is that $X^{-1}=(I_V\otimes S)X$.
The long answer I want to change notation (it should be self-explanatory) and use indices:
$$\begin{aligned}
\langle\eta'|&:=\sum_{q=1}^{\dim V} \overline{b_q}e^q\\
|\eta\rangle&:=\sum_{p=1}^{\dim V}a_pe_p\\
\langle \xi'|&:=\sum_{s=1}^{\dim W}\overline{d_s}f^s\\
|\xi\rangle&:=\sum_{r=1}^{\dim W}c_r f_r.
\end{aligned}$$
Let
$$X=\sum_{i,j=1}^{\dim V}E_{ij}\otimes x_{ij}\text{ and }Y=\sum_{k,l=1}^{\dim W}F_{kl}\otimes y_{kl}.$$
Let us start calculating $\phi:=a$ and $\lambda:=b$.
$$\begin{aligned}
X(|\eta\rangle\otimes I_{A})&=\sum_{i,j=1}^{\dim V}a_je_i\otimes x_{ij}
\\ \implies \phi=(\langle\eta'|\otimes I_A)X(|\eta\rangle\otimes I_{A})&=\sum_{i,j=1}^{\dim V}\overline{b_i}a_j x_{ij},
\end{aligned}$$
using $\mathbb{C}\otimes A\cong A$. Similarly
$$\lambda=\sum_{k,l=1}^{\dim W}\overline{d_k}c_ly_{kl}\implies S(\lambda)=\sum_{k,l=1}^{\dim W}\overline{d_k}c_lS(y_{kl}).$$
So consider
$$h(S(\lambda)\phi)=\sum_{i,j,k,l}\overline{b_i}a_j\overline{d_k}c_lh(S(y_{kl})x_{ij})\qquad(\star).$$
Now let us calculate the right-hand side using
$$Y^{-1}=(I_W\otimes S)Y=\sum_{k,l=1}^{\dim W}F_{kl}\otimes S(y_{kl}).$$
Writing down $R$ (using $R$ rather than $S$) I get:
$$
\begin{aligned}&(\operatorname{id}\otimes h)(Y^{-1}(|\xi\rangle\langle \eta'|\otimes \operatorname{id})X)\\&=(I_W\otimes h)\left(\sum_{k,l=1}^{\dim W}F_{kl}\otimes S(y_{kl})\right)\left[\left(\sum_{r=1}^{\dim W}c_rf_r\right)\left(\sum_{s=1}^{\dim V}\overline{b_q}e^q\right)\otimes I_A\right]\left(\sum_{i,j=1}^{\dim V}E_{ij}\otimes x_{ij}\right).\end{aligned}$$
If you are happy with giving this input $\eta\otimes 1_{\mathbb{C}}$ for $\eta$, we find:
$$\begin{aligned}
R\eta&=\sum_{i,j,k,l}c_l\overline{b_i}a_jf_kh(S(y_{kl})x_{ij})
\\ \implies \langle \xi'|R\eta\rangle&=h(S(\lambda)\phi),
\end{aligned}$$
as required.
