  We prove that the space of pointwise multipliers between two distinct
Musielak--Orlicz spaces is another Musielak-Orlicz space and the function
defining it is given by an appropriately generalized Legendre transform. In
particular, we obtain characterization of pointwise multipliers between Nakano
spaces. We also discuss factorization problem for Musielak-Orlicz spaces and
exhibit some differences between Orlicz and Musielak-Orlicz cases.
