  We determine what appears to be the bare-bones categorical framework for
Poincar\'e-Birkhoff-Witt type theorems about universal enveloping algebras of
various algebraic structures. Our language is that of endofunctors; we
establish that a natural transformation of monads enjoys a
Poincar\'e-Birkhoff-Witt property only if that transformation makes its
codomain a free right module over its domain. We conclude with a number of
applications to show how this unified approach proves various old and new
Poincar\'e-Birkhoff-Witt type theorems. In particular, we prove a PBW type
result for universal enveloping dendriform algebras of pre-Lie algebras,
answering a question of Loday.
