Let π be a group with a unit 1; H is a Hopf π- coalgebra and A is a right π-H-comodule algebra. First, the notion of a two-sided relative （A, H）-Hopf π-comodule is introduced; then it is obtained that Hom A H （M, N） H and HOMA（M, N） are isomorphic as right Hopf π-H-comodules, where Hom A H（M, N） denotes the space of right A-module fight H-comodule morphisms and HOMa （M, N） denotes the rational space of a space Hom A（M, N） of right A-module morphisms. Secondly, the structure theorem of endomorphism algebras of two-sided relative （A, H）-Hopf π--comodules is established; that is, End A H （M）#H and END A（M, N） are isomorphic as fight Hopf π-H-comodules and algebras.
Let A be a bornological quantum group and R a bornological algebra. If R is an essential A-module, then there is a unique extension to M（A）-module with 1x = x. There is a one-to-one corresponding relationship between the actions of A and the coactions of . If R is a Galois object for A, then there exists a faithful δ-invariant functional on R. Moreover,the Galois objects also have modular properties such as algebraic quantum groups. By constructing the comultiplication Δ,counit ε, antipode S and invariant functional φ onR×R, R×R can be considered as a bornological quantum group.
