Partial and unified crossed products are weak crossed products

In Vilaboa, Rodgriguez, and Raposo (2009) the notion of a weak crossed product of an algebra by an object, both living in a monoidal category was presented. Unified crossed products defined in Agore and Militaru (2011) and partial crossed products defined in Muniz et al. (2011) are crossed product structures defined for a Hopf algebra and another object. In this paper we prove that unified crossed products and partial crossed products are particular instances of weak crossed products.