We study the representation theory of the Kazama–Suzuki coset vertex operator superalgebra associated to the pair of a complex simple Lie algebra and its Cartan subalgebra. In the case of type A1, 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.