Invertible skew pairings and crossed products for weak Hopf algebras

In this paper we work with invertible skew pairings for weak bialgebras in a symmetric monoidal category where every idempotent morphism splits. We prove that kind of induces examples distributive laws and therefore they provide wreath products. Also will show define weakly comonoidal mutually inverse pairs and, by the results proved G. Böhm J. Gómez-Torrecillas, obtain products become respect to tensor product coalgebra structure. As an application, Drinfel'd double finite Hopf algebra can be constructed using associated $1$-skew pairing.