  The coalgebraic modelling of alternating automata and of probabilistic
automata has long been obstructed by the absence of distributive laws of the
powerset monad over itself, respectively of the powerset monad over the finite
distribution monad. This can be fixed using the framework of weak distributive
laws. We extend this framework to the case when one of the monads is only a
functor. We provide abstract compositionality results, a generalized
determinization procedure, and systematic soundness of up-to techniques. Along
the way, we apply these results to alternating automata as a motivating
example. Another example is given by probabilistic automata, for which our
results yield soundness of bisimulation up-to convex hull.
