Dear Michael, your result on factorizations of algebra homomorphisms is, in my opinion, new and very nice. It does n o t require the category C to be complete, nor the factorization system to be proper (i.e., it holds even if E-morphisms are not epis). All it requires is a complete factorization system (E,M) , i.e., such that every cone of M-morphisms has a multiple pullback and pullbacks along M-morphisms exist. Then for every monad T the category C^T has the factorization system you desribe: M^T = preimage of M under U, and E^T = the complementary class. The proof is based on Max Kelly's observation that complete systems have M included in Mono. Best regards, Jiri xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 10-Oct-2002 13:00:05 -0300,1719;000000000000-00000000