  In this paper, we investigate morphisms of tautological control systems.
Given a tautological control system $\mathfrak{H}$ on the manifold N and a
mapping $\Phi: M \to N$, we study existence of tautological control system
$\mathfrak{G}$ on the manifold $M$ such that there exists a
trajectory-preserving morphism $(\Phi, \Phi^ #)$ from $\mathfrak{G}$ to
$\mathfrak{H}$. Sufficient conditions are given such that reachability of
$\mathfrak{H}$ implies the reachability of $\mathfrak{G}$. Correspondence
between the notion of lifting ordinary control systems and morphisms of
tautological control systems are examined. We give an application of the above
results to the class of second-order type control systems, where the special
structure of second-order type leads to additional results.
