Enriched duality in double categories: V-categories and V-cocategories

In this work, we explore a double categorical framework for categories of enriched graphs, categories and the newly introduced notion of cocategories. A fundamental goal is to establish an enrichment of V-categories in V-cocategories, which generalizes the so-called Sweedler theory relative to an enrichment of algebras in coalgebras. The language employed is that of V-matrices, and an interplay between the double categorical and bicategorical perspective provides a high-level flexibility for demonstrating essential features of these dual structures. Furthermore, we investigate an enriched fibration structure involving categories of monads and comonads in fibrant double categories a.k.a. proarrow equipments, which leads to natural applications for their materialization as categories and cocategories in the enriched matrices setting.