Sweedler theory for double categories

In this work, we establish certain enrichments of dual algebraic structures in the setting of monoidal double categories. In more detail, we obtain a tensored and cotensored enrichment of monads in comonads, as well as a tensored and cotensored enrichment of modules in comodules, under very general conditions on the surrounding double category. These include `monoidal closedness' and `local presentability', leading classical notions which are here introduced in the double categorical context. Furthermore, we show that in this setting, the actual fibration of modules over monads is itself enriched in the opfibration of comodules over comonads. Applying this abstract double categorical framework to the setting of V-matrices, one directly obtains a many-object generalization of the known enrichment of modules over monoids in comodules over comonoids in a monoidal category V, which was originally induced by Sweedler's measuring k-coalgebras in vector spaces. In the present setting the result involves V-enriched modules of V-categories and V-enriched comodules of V-cocategories.