  We derive correspondences of correlation functions among dual conformal field
theories in two dimensions by developing a "first order formulation" of coset
models. We examine several examples, and the most fundamental one may be a
conjectural equivalence between a coset $(SL(n)_k \otimes
SL(n)_{-1})/SL(n)_{k-1}$ and $\mathfrak{sl}(n)$ Toda field theory with generic
level $k$. Among others, we also complete the derivation of higher rank
FZZ-duality involving a coset $SL(n+1)_k /(SL(n)_{k} \otimes U(1))$, which
could be done only for $n=2,3$ in our previous paper. One obstacle in the
previous work was our poor understanding of a first order formulation of coset
models. In this paper, we establish such a formulation using the BRST
formalism. With our better understanding, we successfully derive correlator
correspondences of dual models including the examples mentioned above. The
dualities may be regarded as conformal field theory realizations of some of the
Gaiotto-Rap\v{c}\'ak dualities of corner vertex operator algebras.
