The structure theorem and duality theorem for endomorphism algebras of weak Hopf group coalgebras

In this paper, we investigate the HOM-functor and state the structure theorem for endomorphism algebras of weak two-sided (H, A)-Hopf pi-modules in order to explore homological algebras for weak Hopf pi-modules, and present the duality theorem for weak group "big" Smash products which extends the result of Menini and Raianu [Morphisms of relative Hopf modules, Smash products and duality, J. Algebra 219 (1999) 547-570] in the setting of weak Hopf group coalgebras.