  The simplicial endofunctor induced by a comonad in some category may underly
a cyclic object in its category of endofunctors. The cyclic symmetry is then
given by a sequence of natural transformations. We write down the commutation
relations the first cyclic operator has to satisfy with the data of the
comonad. If we add a version of quantum Yang Baxter relation and another
relation we actually get a sufficient condition for constructing a sequence of
higher cyclic operators in a canonical fashion. A degenerate case of this
construction comes from so-called trivial symmetry of an additive comonad.
  We also consider weaker versions for paracyclic objects as well as some
connections to the subject of distributive laws.
