  In this dissertation we examine enrichment relations between categories of
dual structure and we sketch an abstract framework where the theory of
fibrations and enriched category theory are appropriately united.
  We initially work in the context of a monoidal category, where we study an
enrichment of the category of monoids in the category of comonoids under
certain assumptions. This is induced by the existence of the universal
measuring comonoid, a notion originally defined by Sweedler in vector spaces.
We then consider the fibred category of modules over arbitrary monoids, and we
establish its enrichment in the opfibred category of comodules over arbitrary
comonoids. This is now exhibited via the existence of the universal measuring
comodule, introduced by Batchelor.
  We then generalize these results to their `many-object' version. In the
setting of the bicategory of V-enriched matrices, we investigate an enrichment
of V-categories in V-cocategories as well as of V-modules in V-comodules. This
part constitutes the core of this treatment, and the theory of fibrations and
adjunctions between them plays a central role in the development. The newly
constructed categories are described in detail, and they appropriately fit in a
picture of duality, enrichment and fibrations as in the previous case.
  Finally, we introduce the concept of an enriched fibration, aimed to provide
a formal description for the above examples. Related work in this direction,
though from a different perspective and with dissimilar outcomes, has been
realized by Shulman. We also discuss an abstraction of this picture in the
environment of double categories, concerning categories of monoids and modules
therein.
