  Every monoidal functor G: C --> M has a canonical factorization through the
category of bimodules over some monoid R in M such that the factor U: C -->_R
M_R is strongly unital. Using this result and the characterization of the
forgetful functors M_A -->_R M_R of bialgebroids A over R given by Schauenburg
together with their bimonad description given by the author recently here we
characterize the "long" forgetful functors M_A -->_R M_R --> M of both
bialgebroids and weak bialgebras.
