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