  The universal (co)acting bi/Hopf algebras introduced by Yu.\,I.~Manin,
M.~Sweedler and D.~Tambara, the universal Hopf algebra of a given (co)module
structure, as well as the universal group of a grading, introduced by J.~Patera
and H.~Zassenhaus, find their applications in the classification of quantum
symmetries. Typically, universal (co)acting objects are defined as initial or
terminal in the corresponding categories and, as such, they do not always
exist. In order to ensure their existence, we introduce the support of a given
object, which generalizes the support of a grading and is used to restrict the
class of objects under consideration. The existence problems for universal
objects are formulated and studied in a purely categorical manner by seeing
them as particular cases of the lifting problem for a locally initial object.
We prove the existence of a lifting and, consequently, of the universal
(co)acting objects under some assumptions on the base (braided or symmetric
monoidal) category. In contrast to existing constructions, our approach is
self-dual in the sense that we can use the same proof to obtain the existence
of universal actions and coactions. In particular, when the base category is
the category of vector spaces over a field, the category of sets or their
duals, we recover known existence results for the aforementioned universal
objects. The proposed approach allows us to apply our results not only to the
classical categories of sets and vectors spaces and their duals but also to
(co)modules over bi/Hopf algebras, differential graded vector spaces, $G$-sets
and graded sets.
