  We study the existence of universal measuring comonoids $P(A,B)$ for a pair
of monoids $A$, $B$ in a braided monoidal closed category, and the associated
enrichment of a category of monoids over the monoidal category of comonoids. In
symmetric categories, we show that if $A$ is a bimonoid and $B$ is a
commutative monoid, then $P(A,B)$ is a bimonoid; in addition, if $A$ is a
cocommutative Hopf monoid then $P(A,B)$ always is Hopf. If $A$ is a Hopf
monoid, not necessarily cocommutative, then $P(A,B)$ is Hopf if the fundamental
theorem of comodules holds; to prove this we give an alternative description of
the dualizable $P(A,B)$-comodules and use the theory of Hopf (co)monads. We
explore the examples of universal measuring comonoids in vector spaces and
graded spaces.
