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

Endofunctors and Poincar\'e-Birkhoff-Witt theorems