  In this work, we continue the investigation of certain enrichments of dual
algebraic structures in monoidal double categories, that was initiated in
[Vas19]. First, we re-visit monads and comonads in double categories and
establish a tensored and cotensored enrichment of the former in the latter,
under general conditions. These include monoidal closedness and local
presentability of the double category, notions that are proposed as tools
required for our main results, but are of interest in their own right. The
natural next step involves categories of the newly introduced modules for
monads and comodules for comonads in double categories. After we study their
main categorical properties, we establish a tensored and cotensored enrichment
of modules in comodules, as well as an enriched fibration structure that
involves (co)modules over (co)monads in double categories. Applying this
abstract double categorical framework to the setting of V-matrices produces an
enrichment of the category of V-enriched modules (fibred over V-categories) in
V-enriched comodules (opfibred over V-cocategories), which is the many-object
generalization of the respective result for modules (over algebras) and
comodules (over coalgebras) in monoidal categories.
