Does the antipode leave Peter-Weyl blocks invariant.

Question

Let $\mathcal U_q(\mathfrak{g})$ be a algebraic quantum group over $\mathbf{Q}(q)$, and $A_q(G)$ be its linear dual of matrix entires. Suppose I have a matrix entry $c_{f,v}\in L^r(\lambda)\otimes L(\lambda)$ for some dominant weight $\lambda$. Then, for instance in Jantzen's book on quantum groups chapter 7.11, it is claimed that $c_{f,v}\circ S=c_{v,f}$ where $v$ is identified with its image $L(\lambda)\to (L(\lambda)^\ast)^\ast$. As highest weight modules $L(\lambda)$ and $(L(\lambda)^\ast)^\ast$ are isomorphic, is this enough to conclude that $c_{f,v}\circ S\in L^r(\lambda)\otimes L(\lambda)$?