The monoidalEilenberg–Moore construction and bialgebroids

Strong monoidalfunctors with left adjoints determine, in a universal way, monoidsT in the category of opmonoidalendofunctorson . Treating such opmonoidal monads as abstract “quantum groupoids” we derive Tannaka duality between right adjoint strong monoidalfunctors and opmonoidal monads. Bialgebroids, i.e., Takeuchiʹs ×R-bialgebras, appear as the special case when T has also a right adjoint. Streetʹs 2-category of monads then leads to a natural definition of the 2-category of bialgebroids.