  The notion of proof-net category defined in this paper is closely related to
graphs implicit in proof nets for the multiplicative fragment without constant
propositions of linear logic. Analogous graphs occur in Kelly's and Mac Lane's
coherence theorem for symmetric monoidal closed categories. A coherence theorem
with respect to these graphs is proved for proof-net categories. Such a
coherence theorem is also proved in the presence of arrows corresponding to the
mix principle of linear logic. The notion of proof-net category catches the
unit free fragment of the notion of star-autonomous category, a special kind of
symmetric monoidal closed category.
