Projections and Yetter–Drinfel'd modules over Hopf (co)quasigroups

In this paper we introduce the notions of Yetter–Drinfel'd module and strong projection over a Hopf quasigroup H. If the antipode of H   is invertible we show that the category of left–left Yetter–Drinfel'd modules YDHH is braided, and there exists a categorical equivalence between the category of Hopf quasigroups in YDHH and the category of strong Hopf algebra projections associated to H.