Correspondences of Categories for Subregular $${{\mathcal {W}}}$$-Algebras and Principal $${\mathcal {W}}$$-Superalgebras

Based on the Kazama-Suzuki type coset construction and its inverse between subregular $\mathcal{W}$-algebras for $\mathfrak{sl}_n$ principal $\mathcal{W}$-superalgebras $\mathfrak{sl}_{1|n}$, we prove weight-wise linear equivalences of their representation categories. Our main results are then improvements these correspondences incorporating monoidal structures. Firstly, in rational case, obtain classification simple modules fusion rules via current extensions from Heisenberg cosets. Secondly, beyond use certain kernel VOAs together with relative semi-infinite cohomology functors to get categories $\mathfrak{sl}_{1|n}$ vice versa. We study particular isomorphisms superspaces logarithmic intertwining operators. As a corollary, sense case as well.