Biproduct dagger categories serve as models for natural language. They link the extensional models of predicate calculus with the intensional models of quantum logic. The morphisms representing the extensional meanings of a grammatical string are translated to projectors representing the intensional meanings such that truth is preserved. Pregroup grammars serve as the tool that transforms a grammatical string into a morphism. The chosen linguistic examples concern negation, relative noun phrases, comprehension and quantifiers.