It is shown that the multiplicative monoids of Temperley-Lieb algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices. This selfadjunction underlies the orthogonal group case of Brauer's representation of the Brauer centralizer algebras.

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.

This paper presents a proof-theoretical approach to the question "What is a logical constant?" This approach starts with the assumption that logic is the science of formal deductions, and that basic formal deductions are structural deductions, i.e. deductions independent of any constant of the language to which the premises and conclusions belong. Logical constants, on which the remaining formal deductions are dependent, may be said to serve as "punctuation marks" for some structural features of deductions; this punctuation function, exhibited in equivalences which amount to analyses of logical constants, is taken as a criterion for being a logical constant. The paper presents an account of philosophical analysis which covers the proposed analyses of logical constants. Some related assumptions concerning logic are also considered. In particular, since a logical system is completely determined by its structural deductions, alternative logical systems arise by changing structural deductions while having constants with the same punctuation function. Some other approaches to the question "What is a logical constant?", grammatical, model-theoretical, and proof-theoretical, are briefly considered.

This paper gives a self-contained and complete proof of the isomorphism of freely generated monoids extracted from Temperley-Lieb algebras with monoids made of Kauffman's diagrams.