  We study integrals of Hopf monoids in duoidal endohom categories of naturally
Frobenius map monoidales in monoidal bicategories. We prove two Maschke type
theorems, relating the separability of the underlying monoid and comonoid,
respectively, to the existence of normalized integrals. It covers the examples
provided by Hopf monoids in braided monoidal categories, weak Hopf algebras,
Hopf algebroids over central base algebras, Hopf monads on autonomous monoidal
categories and Hopf categories.
