  This paper is concerned with the theory of cup-products in Hopf-type cyclic
cohomology of algebras and coalgebras. Here we give detailed proofs of the
statements, announced in our previous paper. We show that the cyclic cohomology
of a coalgebra can be obtained from a construction involving noncommutative
Weil algebra. Then we use a generalization of Quillen and Crainic's
construction to define the cup-product. We discuss the relation of the
introduced cup-product and $S$-operations on cyclic cohomology. After this we
describe the relation of this type of product and bivariant cyclic cohomology.
In the last section we briefly discuss the relation of our constructions with
that of Khalkhali and Rangipour.
