Factorization systems as Eilenberg-Moore algebras

AbstractA functorial notion of factorization system is introduced and shown to coincide with the appropriate 2-categorical notion of algebra, with respect to the monad on Cat which assigns to a category K its morphism category K2. Quite surprisingly, it is also equivalent to the common notion of (E, M)-factorization system with the usual diagonalization property; in brief: functorial choice of ‘diagonals’ necessarily yields their uniqueness