  Riehl and Verity have introduced an "$\infty$-cosmic" framework in which they
redevelop the category theory of $\infty$-categories using 2-categorical
arguments. In this paper, we begin with a self-contained review of the parts of
their theory needed to discuss adjunctions and monadicity. This is applied in
order to extend to the $\infty$-categorical context the classical criterion for
fully faithfulness of the comparison functor induced by an adjunction. We
discuss the relation with previous work in the literature---which primarily
uses model-categorical techniques---and indicate applications to descent
theory.
