On ultraweak continuity

Let A be a C*-algebra, f be a representation of A, F be the universal representation of A, and g=foF^(-1). For an ultraweakly continuous linear functional w on f(A), wog is bounded and hence according to a well-known theorem, is ultraweakly continuous linear functional on F(A).
My question: it results that g is ultraweakly continuous. How is its proof?
Thank you for consideration.