  Brugui\`eres, Lack and Virelizier have recently obtained a vast
generalization of Sweedler's Fundamental Theorem of Hopf modules, in which the
role of the Hopf algebra is played by a bimonad. We present an extension of
this result which involves, in addition to the bimonad, a comodule-monad and a
algebra-comonoid over it. As an application we obtain a generalization of
another classical theorem from the Hopf algebra literature, due to Schneider,
which itself is an extension of Sweedler's result (to the setting of Hopf
Galois extensions).
