  This is the first one in a series of two papers on the continuation of our
study in cup products in Hopf cyclic cohomology. In this note we construct
cyclic cocycles of algebras out of Hopf cyclic cocycles of algebras and
coalgebras. In the next paper we consider producing Hopf cyclic cocycle from
"equivariant" Hopf cyclic cocycles. Our approach in both situations is based on
(co)cyclic modules and bi(co)cyclic modules together with Eilenberg-Zilber
theorem which is different from the old definition of cup products defined via
traces and cotraces on DG algebras and coalgebras.
