  Hopf monads generalise Hopf algebras. They clarify several aspects of the
theory of Hopf algebras and capture several related structures such as weak
Hopf algebras and Hopf algebroids. However, important parts of Hopf algebra
theory are not reached by Hopf monads, most noticeably Drinfeld's quasi-Hopf
algebras. In this paper we introduce a generalisation of Hopf monads, that we
call slack Hopf monads. This generalisation retains a clean theory and is
flexible enough to encompass quasi-Hopf algebras as examples. A slack Hopf
monad is a colax magma monad $T$ on a magma category $\mathcal{C}$ such that
the forgetful functor $U^T : \mathcal{C}^T \rightarrow \mathcal{C}$ `slackly'
preserves internal Homs. We give a number of different descriptions of slack
Hopf monads, and study special cases such as slack Hopf monads on cartesian
categories and $k$-linear exact slack Hopf monads on $\mathrm{Vect}_k$, that is
comagma algebras such that a modified fusion operator is invertible. In
particular, we characterise quasi-Hopf algebras in terms of slackness.
