In this chapter an analysis is carried out which is analogous to, but more general than that in Chap. 4. The category of vector spaces in Chap. 4 is replaced by the category of bimodules over some algebra B; or, isomorphically, the category of left modules over B ⊗ Bop. Those endofunctors on it are considered which are induced, as in Example 2.5 4, by the B ⊗ Bop-module tensor product with a fixed B ⊗ Bop-bimodule A. The monad structures on this functor A⊗B⊗Bop− are related to the algebra homomorphisms B ⊗ Bop → A. The monoidal structure of the category of B-bimodules is explained in Example 3.2 5. The opmonoidal structures with respect to it on the endofunctor A⊗B⊗Bop−, are related to the so-called B|B-coring structures on A. This results in a bijection between the bimonads with underlying functor A⊗B⊗Bop− on the category of B-bimodules; and the bialgebroids over the base algebra B. The bijection is shown to restrict to Hopf monads on one hand; and Hopf algebroids on the other hand.