Abstract proof of the Kleisli–Ctx equivalence for polynomial monads

Prove Theorem thm:pra.monads.transpose establishing that for any monad T whose underlying functor is polynomial, with its transposed contextad C, there is an identity-on-objects isomorphism between the Kleisli category Kl(T) and the category of contexful arrows of Ctx(C). More generally, develop the abstract proof via the framework of parametric right adjoint monads to show their transposability to dependently graded comonads.

Background

Earlier in the paper, the authors motivate a general transposition principle: many monads modeling effects, particularly those with polynomial underlying functors, should correspond to contextads whose Ctx construction reproduces the Kleisli category. They sketch the correspondence in detail for polynomial monads in the appendix.

They state Theorem thm:pra.monads.transpose asserting an equivalence between Kl(T) and the category of contexful arrows arising from the transposed contextad. However, they explicitly leave this theorem without proof and indicate their intent to provide an abstract proof based on parametric right adjoint monads, which would yield a broader transposability result.