Weak C-cleft extensions, weak entwining structures and weak Hopf algebras

We formulate the concept of weak cleft extension for a weak entwining structure in a braided monoidal category C with equalizers and coequalizers. We prove that if A is a weak C-cleft extension, then there is an isomorphism of algebras between A and a subobject of the tensor product of AC and C where AC is a subalgebra of A. Also, we prove the corresponding dual results and linking the information of this two parts we obtain a general property for a pair morphisms and of algebras and coalgebras satisfying certain conditions. Finally, as particular instances, we get the results of Fernández and Rodríguez, the theorems of Radford, Majid and Bespalov (in the case of Hopf algebras with projection) and the ones obtained by Alonso and González for weak Hopf algebras living in a symmetric category with split idempotents, for example, the weak theorem of Blattner, Cohen and Montgomery for weak Hopf algebras with coalgebra splitting is one of them.