  We develop a theory which unifies the universal (co)acting bi/Hopf algebras
as studied by Sweedler, Manin and Tambara with the recently introduced
\cite{AGV1} bi/Hopf-algebras that are universal among all support equivalent
(co)acting bi/Hopf algebras. Our approach uses vector spaces endowed with a
family of linear maps between tensor powers of $A$, called $\Omega$-algebras.
This allows us to treat algebras, coalgebras, braided vector spaces and many
other structures in a unified way. We study $V$-universal measuring coalgebras
and $V$-universal comeasuring algebras between $\Omega$-algebras $A$ and $B$,
relative to a fixed subspace $V$ of $\Vect(A,B)$. By considering the case
$A=B$, we derive the notion of a $V$-universal (co)acting bialgebra (and Hopf
algebra) for a given algebra $A$. In particular, this leads to a refinement of
the existence conditions for the Manin--Tambara universal coacting bi/Hopf
algebras. We establish an isomorphism between the $V$-universal acting bi/Hopf
algebra and the finite dual of the $V$-universal coacting bi/Hopf algebra under
certain conditions on $V$ in terms of the finite topology on $\End_F(A)$.
