Self-adjunctions and matrices

It is shown that the multiplicative monoids of Temperley–Lieb algebras 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 functoradjoint to itself is based on the Kronecker product of matrices. This self-adjunction underlies the orthogonal group case of Brauerʹs representation of the Brauer centralizer algebras.