In this paper, we introduce the notions of \(\phi \) - \(\textrm{WAP}\) -biprojectivity and \(\phi \) - \(\textrm{WAP}\) -virtual diagonal for the enveloping dual Banach algebras F(A) and we study the relation between these notions. We then show that F(A) is \(\phi \) - \(\textrm{WAP}\) -biprojective if A is \(\phi \) -biprojective or \(\phi \) -Johnson contractible. Examples are provided to demonstrate that \(\phi \) - \(\textrm{WAP}\) -biprojectivity is distinct from \(\phi \) -biprojectivity and from \(\textrm{WAP}\) -biprojectivity. Finally, we define the notion of \(\phi \) -Connes biprojectivity of dual Banach algebras and find some relations between the new notions of \(\phi \) - \(\textrm{WAP}\) -biprojectivity, \(\phi \) -Connes biprojectivity and some concepts already known.
