This paper is a companion to another paper where 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 underlies the orthogonal group case of Brauer's representation of the Brauer centralizer algebras. The present paper provides detailed proofs of results on the presentation of various monoids of diagrams by generators and relations, on which the other paper depends.