  We study the existence of universal measuring comodules Q(M,N) for a pair of
modules M,N in a braided monoidal closed category, and the associated
enrichment of the global category of modules over the monoidal global category
of comodules. In the process, we use results for general fibred adjunctions
encompassing the fibred structure of modules over monoids and the opfibred
structure of comodules over comonoids. We also explore applications to the
theory of Hopf modules.
