Where does the unitarity structure of $U_q(\mathfrak{sl}_2)$ come from?

Question

It is known that when $q$ is the root of unity, the representation of the quantum group $U_q(\mathfrak{sl}_2)$ is a unitary modular tensor category. However, if we want it to have the dagger structure, we not only need to have a dagger structure on $\mathbf{Vec}$, but if we do not have dagger structure on the Hopf algebra level, how can it be satisfied? If $S$ intertwines with the representation $(V,\rho_V)$ and $(W,\rho_W)$, i.e., $$ S \rho_V(x) = \rho_W(x) S \,, $$ then the dagger of $S$ should also intertwine with the representation, i.e., $$ \rho_V(x) S^\dagger = S^\dagger \rho_W(x) \,. $$ But how can it be satisfied?