On the monad of proper factorisation systems in categories

It is known that factorisation systems in categories can be viewed as unitary pseudo-algebras for the monad image, in Cat. We show in this note that an analogous fact holds for proper (i.e., epi-mono) factorisation systems and a suitable quotient of the former monad, deriving from a construct introduced by Freyd for stable homotopy. Some similarities of image with the structure of the path endofunctor of topological spaces are considered.