Formulate the general selection-category framework for the monad–comonad distributive-law case

Determine an appropriate categorical formulation that extends the selection category construction—where a category is obtained from a polynomial functor p: Set → Set and a small category C via the comonad Lan{p ∘ C}{p}—to the full generality of Theorem thm.monad_comonad_dist: given accessible categories C and D, an accessible functor p: C → D, an accessible monad t on C, an accessible comonad k on C, and a distributive law α: t ∘ k → k ∘ t, establish how the comonad Lan{p ∘ k}{p ∘ t} on D should be organized and presented in a manner analogous to selection categories.

Background

Selection categories, introduced in this paper, arise from the comonads Lan{p ∘ C}{p} where p is a polynomial functor on Set and C is a small category. The paper proves well-behavedness properties (including a P-enriched functoriality and preservation of certain factorization properties) for this construction.

Earlier, Theorem thm.monad_comonad_dist shows that in a much broader setting—when a monad t and a comonad k on C admit a distributive law α: t ∘ k → k ∘ t—the left Kan extension Lan{p ∘ k}{p ∘ t} carries a comonad structure on D. The authors indicate that a “more general story” should exist that parallels the selection-category viewpoint in this broader context, but they do not provide it here and explicitly state that finding the right way to formulate it is an open question.