  The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in
the simplicial category, are submonoids of monoids one finds in Temperley-Lieb
algebras, and as the monoids of Temperley-Lieb algebras are linked to
situations where an endofunctor is adjoint to itself, so the monoids of
simplicial endomorphisms are linked to arbitrary adjoint situations. This link
is established through diagrams of the kind found in Temperley-Lieb algebras.
Results about these matters, which were previously prefigured up to a point,
are here surveyed and reworked. A presentation of monoids of simplicial
endomorphisms by generators and relations has been given a long time ago. Here
a closely related presentation is given, with completeness proved in a new and
self-contained manner.
