Boardman-Vogt tensor product and wreath product of operadic categories

We introduce the wreath product for a class of operadic categories and use it to construct an explicit isomorphism between the Boardman-Vogt tensor product of two colored operads in Set and an operad induced by the wreath product of operadic Grothendieck constructions of the respective operads. We also describe how the wreath product of operadic categories is related to the wreath product of categories introduced by C. Berger to characterize n-fold loop spaces.