Abstract: To further investigate the sensitivity of non-autonomous dynamical systems, the initial sensitivity and n-sensitivity in non-autonomous dynamical systems are studied through Furstenberg family theory. The equivalent condition for the non-autonomous dynamical system to be sensitive is given, and it is demonstrated that \mathcalF-n-sensitivity can be enhanced by factor mapping. It is proven that for non-autonomous dynamical systems(X, f_{1, ∞}), where X is a locally connected space and \mathcalF is a Furstenberg family, if it is \mathcalF-sensitive, then the system is completely weakly \mathcalF initial value sensitive.