  We introduce and study bimeasurings from pairs of bialgebras to algebras. It
is shown that the universal bimeasuring bialgebra construction, which arises
from Sweedler's universal measuring coalgebra construction and generalizes the
finite dual, gives rise to a contravariant functor on the category of
bialgebras adjoint to itself. An interpretation of bimeasurings as algebras in
the category of Hopf modules is considered.
