  We derive a uniqueness result for non-Cartesian composition of systems in a
large class of process theories, with important implications for quantum theory
and linguistics. Specifically, we consider theories of wavefunctions valued in
commutative involutive semirings -- as modelled by categories of free
finite-dimensional modules -- and we prove that the only bilinear
compact-closed symmetric monoidal structure is the canonical one (up to linear
monoidal equivalence). Our results apply to conventional quantum theory and
other toy theories of interest in the literature, such as real quantum theory,
relational quantum theory, hyperbolic quantum theory and modal quantum theory.
In computational linguistics they imply that linear models for categorical
compositional distributional semantics (DisCoCat) -- such as vector spaces,
sets and relations, and sets and histograms -- admit an (essentially) unique
compatible pregroup grammar.
