Measuring Comodules and Enrichment

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.