A family of unbounded linear operators $(S(t))_{t\geq0}$ in the Banach space $(E,\|\cdot\|)$ which satisfies the composition law for an integrated $C$-semigroup on a domain $D\subset E$ is introduced and investigated. The Banach spaces $(E_\omega,\|\cdot\|_\omega)$, $\omega>0$, are used for the construction of a family of infinitesimal generators $A^\omega$, $\omega>0$ which determine an operator $A$ called the infinitesimal generator of $(S(t))_{t\geq0}$.