ABSTRACT
Every monoidal functor G : C → ℳ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187629/ac445a68-1635-4dfb-97a7-7653d5efb83a/content/inq_chapter18_291_1.tif"/> has a canonical factorization through the category Rℳ R of bimodules in ℳ over some monoid R in ℳ in which the factor U : C → R ℳ R https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187629/ac445a68-1635-4dfb-97a7-7653d5efb83a/content/inq_chapter18_291_2.tif"/> 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 [15] together with their bimonad description given by the author in [18] here we characterize the ”long” forgetful functors ℳ A → R ℳ R → ℳ of both bialgebroids and weak bialgebras.
