Dear Paul, Let us forget about the monoidal unit and fix (a). You define a strict semi? monoidal structure on S letting x*y = x. Then your pair (F, beta) is a lax monoidal functor M --> S. Your condition (a) is the hexagon of consistency with alpha, which here reduces to: F(p) = F(p) || | F(p) F(p*a) | | F(p*(a*b) --> F((p*a)*b) (a single | stands for a downward beta) Now, to fix also (b), I guess you should add to S a new object which is a strict identity for the tensor and work out things. However, if your problem is only about terminology and you do not want to use the tensor on S in the sequel (eg to compose F with other monoidal functors), you might not bother about that. Best regards Marco