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In this work, we explore a double categorical framework for categories ofenriched graphs, categories and the newly introduced notion of cocategories. Afundamental goal is to establish an enrichment of V-categories inV-cocategories, which generalizes the so-called Sweedler theory relatively toan enrichment of algebras in coalgebras. The language employed is that ofV-matrices, and an interplay between the double categorical and bicategoricalperspective provides a high-level flexibility for demonstrating essentialfeatures of these dual structures. Furthermore, we investigate an enrichedfibration structure involving categories of monads and comonads in fibrantdouble categories a.k.a. proarrow equipments, which leads to naturalapplications for their materialization as categories and cocategories in theenriched matrices setting.

Enriched Duality in Double Categories: V-categories and V-cocategories