  We introduce the notion of an accessible $\infty$-cosmos and prove that these
include the basic examples of $\infty$-cosmoi and are stable under the main
constructions. A consequence is that the vast majority of known examples of
$\infty$-cosmoi are accessible. By the adjoint functor theorem for
homotopically enriched categories which we proved in an earlier paper, joint
with Lukas Vokrinek, it follows, for instance, that all such $\infty$-cosmoi
have flexibly weighted homotopy colimits.
