  In general, universal (co)measuring (co)monoids and universal (co)acting
bi/Hopf monoids, which prove to be a useful tool in the classification of
quantum symmetries, do not always exist. In order to ensure their existence,
the support of a given object was recently introduced in \cite{AGV3} and used
to restrict the class of objects considered when defining universal (co)acting
objects. It is well-known that, in contrast with the universal coacting Hopf
algebra, for actions on algebras over a field it is usually difficult to
describe the universal acting Hopf algebra explicitly and this turns the
duality theorem into an important investigation tool. In the present paper we
establish duality results for universal (co)measuring (co)monoids and universal
(co)acting bi/Hopf monoids in pre-rigid braided monoidal categories $\mathcal
C$. In addition, when the base category $\mathcal C$ is closed monoidal, we
provide a convenient uniform approach to the aforementioned universal objects
in terms of the cosupports, which in this case become subobjects of internal
hom-objects. In order to explain our constructions, we use the language of
locally initial objects. Known results from the literature are recovered when
the base category is the category of vector spaces over a field. New cases
where our results can be applied are explored, including categories of
(co)modules over (co)quasitriangular Hopf algebras, Yetter-Drinfeld modules and
dg-vector spaces.
