  In this paper, we prove the equivalence between two braided tensor
categories. On the one hand, we consider the category of modules of the
Virasoro vertex operator algebra with a generic central charge (generic
Virasoro VOA) generated by those simple modules lying in the first row of the
Kac table. On the other hand, we take the category of finite-dimensional type I
modules of the quantum group $\mathcal{U}_{q} (\mathfrak{sl}_{2})$ with $q$
determined by the central charge. This is a continuation of our previous work
in which we examined intertwining operators for the generic Virasoro VOA in
detail. Our strategy to show the categorical equivalence is to take those
results as input and directly compare the structures of the tensor categories.
Therefore, we are to execute the most elementary proof of categorical
equivalence. We also study the category of $C_{1}$-cofinite modules of the
generic Virasoro VOA. We show that it is equivalent to the category of
finite-dimensional type I modules of $\mathcal{U}_{q}
(\mathfrak{sl}_{2})\otimes \mathcal{U}_{\tilde{q}} (\mathfrak{sl}_{2})$ as a
tensor category, where $q$ and $\tilde{q}$ are again related to the central
charge.
