Determine the relationship between Hermida’s fibrational modalities and the Kan-extension-based two-dimensional Kripke semantics

Determine the precise relationship between Hermida’s fibrational account of relational modalities—where modalities arise as compositions of adjoints in a fibrational setting—and the Kan extension-based adjunction on presheaf categories that yields the two-dimensional Kripke semantics for modal logic developed using endoprofunctors on a Cauchy-complete base category. Clarify whether and how the constructions correspond, and under what conditions they coincide or differ.

Background

Hermida presents a fibrational account of relational modalities, deriving both diamond and box modalities canonically as extensions of predicate logic to relations, with modalities arising as compositions of adjoints. In that setting, the left adjoint to box (the black diamond) briefly appears as well.

In contrast, the present paper derives intuitionistic modal operators on presheaf categories via Kan extension from an endoprofunctor on a Cauchy-complete base category, thereby connecting two-dimensional Kripke semantics and categorical semantics.

The author notes that Kan extension does not explicitly appear in Hermida’s treatment, leaving the precise comparison between the two frameworks unresolved.