  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 relatively 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.
