  We give a new criterion guaranteeing existence of model structures
left-induced along a functor admitting both adjoints. This works under the
hypothesis that the functor induces idempotent adjunctions at the homotopy
category level. As an application, we construct new model structures on cubical
sets, prederivators, marked simplicial sets and simplicial spaces modeling
$\infty$-categories and $\infty$-groupoids.
