  Let A be a Hopf algebra and H a coalgebra. We shall describe and classify up
to an isomorphism all Hopf algebras E that factorize through A and H: that is E
is a Hopf algebra such that A is a Hopf subalgebra of E, H is a subcoalgebra in
E with 1_{E} \in H and the multiplication map $A\otimes H \to E$ is bijective.
The tool we use is a new product, we call it the unified product, in the
construction of which A and H are connected by three coalgebra maps: two
actions and a generalized cocycle. Both the crossed product of an Hopf algebra
acting on an algebra and the bicrossed product of two Hopf algebras are special
cases of the unified product. A Hopf algebra E factorizes through A and H if
and only if E is isomorphic to a unified product of A and H. All such Hopf
algebras E are classified up to an isomorphism that stabilizes A and H by a
Schreier type classification theorem. A coalgebra version of lazy 1-cocycles as
defined by Bichon and Kassel plays the key role in the classification theorem.
