  In this paper, we associate a dual quasi-bialgebra, called bosonization, to
every dual quasi-bialgebra $H$ and every bialgebra $R$ in the category of
Yetter-Drinfeld modules over $H$. Then, using the fundamental theorem, we
characterize as bosonizations the dual quasi-bialgebras with a projection onto
a dual quasi-bialgebra with a preantipode. As an application we investigate the
structure of the graded coalgebra $grA$ associated to a dual quasi-bialgebra
$A$ with the dual Chevalley property (e.g. $A$ is pointed).
