Endofunctors and Poincar\'e-Birkhoff-Witt theorems

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.