  In this paper, we introduce a cofibrant simplicial category that we call the
free homotopy coherent adjunction and characterize its n-arrows using a
graphical calculus that we develop here. The hom-spaces are appropriately
fibrant, indeed are nerves of categories, which indicates that all of the
expected coherence equations in each dimension are present. To justify our
terminology, we prove that any adjunction of quasi-categories extends to a
homotopy coherent adjunction and furthermore that these extensions are
homotopically unique in the sense that the relevant spaces of extensions are
contractible Kan complexes.
  We extract several simplicial functors from the free homotopy coherent
adjunction and show that quasi-categories are closed under weighted limits with
these weights. These weighted limits are used to define the homotopy coherent
monadic adjunction associated to a homotopy coherent monad. We show that each
vertex in the quasi-category of algebras for a homotopy coherent monad is a
codescent object of a canonical diagram of free algebras. To conclude, we prove
the quasi-categorical monadicity theorem, describing conditions under which the
canonical comparison functor from a homotopy coherent adjunction to the
associated monadic adjunction is an equivalence of quasi-categories. Our proofs
reveal that a mild variant of Beck's argument is "all in the weights" - much of
it independent of the quasi-categorical context.
