  As a development of [2] and [3], we construct a "VN-bialgebra" in Vect_k for
each k-linear split-semigroupal functor from a suitable monoidal category C to
Vect_k. The main aim here is to avoid the customary compactness assumptions on
generators of the domain category C (cf. [3]). Please note that the
VN-bialgebras in Vect_k defined here are not necessarily von Neumann regular as
k-algebras in the usual sense, the prefix "VN-" coming from the Set-based case
of von Neumann regular semigroups.
