  In previous work we proved that, for categories of free finite-dimensional
modules over a commutative semiring, linear compact-closed symmetric monoidal
structure is a property, rather than a structure. That is, if there is such a
structure, then it is uniquely defined (up to monoidal equivalence). Here we
provide a novel unifying category-theoretic notion of symmetric monoidal
structure with local character, which we prove to be a property for a much
broader spectrum of categorical examples, including the infinite-dimensional
case of relations over a quantale and the non-free case of finitely generated
modules over a principal ideal domain.
