The biproduct bialgebra has been generalized to generalized biproduct bialgebra B

D in [5]. Let (D, B) be an admissible pair and let D be a bialgebra. We show that if generalized biproduct bialgebra B

D is a Hopf algebra with antipode s, then D is a Hopf algebra and the identity

has an inverse in the convolution algebra

(B, B). We show that if D is a Hopf algebra with antipode

(B, B) is an inverse of

then B

D is a Hopf algebra with antipode s described by

. We show that the mapping system

are the canonical inclusions,

are the canonical coalgebra projections) characterizes B

D. These generalize the corresponding results in [6].