Dear All Let (L1,R1) and (L2,R2) be two weak factorization systems (by weak, it is usually meant that the diagonal map is not unique) such that L1 \subset L2. Example of such a thing: in a locally presentable category, take two sets of maps I1 and I2 with I1\subset I2, and let (L1,R1)=(cof(I1),inj(I1)) and (L2,R2)=(cof(I2),inj(I2)). Two such wfs are called cofibrantly generated. Now come back to (L1,R1) and (L2,R2) with L1 \subset L2. Question: if (L2,R2) is cofibrantly generated, is (L1,R1) cofibrantly generated ? if we work in a locally presentable category, can one say something under Vopenka's principle ? The only example I have in mind is the wfs (Serre cofibration, trivial Serre fibration) which is cofibrantly generated by {S^{n-1}->D^n} and (Hurewicz cofibration, Hurewicz fibration which are homotopy equivalences) which is probably not cofibrantly generated. Unfortunately, the inclusion is in the wrong direction : {Serre cofibration}\subset {Hurewicz cofibration} since {Hurewicz fibration which are homotopy equivalences} \subset {trivial Serre fibration} so that does not give a counterexample. Thanks in advance. pg.