  We study the representation theory of the Kazama-Suzuki coset vertex operator
superalgebra associated with the pair of a complex simple Lie algebra and its
Cartan subalgebra. In the case of type $A_{1}$, B.L. Feigin, A.M. Semikhatov,
and I.Yu. Tipunin introduced another coset construction, which is "inverse" of
the Kazama-Suzuki coset construction. In this paper we generalize the latter
coset construction to arbitrary type and establish a categorical equivalence
between the categories of certain modules over an affine vertex operator
algebra and the corresponding Kazama-Suzuki coset vertex operator superalgebra.
Moreover, when the affine vertex operator algebra is regular, we prove that the
corresponding Kazama-Suzuki coset vertex operator superalgebra is also regular
and the category of its ordinary modules carries a braided monoidal category
structure by the theory of vertex tensor categories.
