Most recent papers define cosets of $V_k(g)$ by $V_k(h)$, where $h\subset g$ - some affine (super-)Lie algebras, as a cohomology of a complex

$$V_k(g)\otimes V_{-k}(h)\otimes ghosts$$

but I'm failing to understand why is this isomorphic to usual definition, as in "elements of $V_k(g)$ annihilated by vertex operators of $V_k(h)$".

In my searches for a proof, I only managed to find some physics papers from 90's were they check that conformal vector is the same in cohomology. So I would appreciate if someone points me towards a reference with something more rigorous.

I also wonder if there's a more general analogue for a generic coset of a VOA $V$ with respect to it's subset $W$.