  We study the product formula $(fg)(A) = f(A)g(A)$ in the framework of
(unbounded) functional calculus of sectorial operators $A$. We give an abstract
result, and, as corollaries, we obtain new product formulas for the holomorphic
functional calculus, an extended Stieltjes functional calculus and an extended
Hille-Phillips functional calculus. Our results generalise previous work of
Hirsch, Martinez and Sanz, and Schilling.
