In this paper, the output synchronization problem for nonlinear heterogeneous multi-agent systems is studied in a switching network of time-varying directed graphs. The scenario under consideration is complicated in the sense that none of the graphs is assumed to be connected (containing a spanning tree). Within a two-step framework for output synchronization of nonlinear heterogeneous multi-agent systems, the main technical challenge lies in the existence of switching strategies under which a class of switched unstable subsystems subject to external perturbation can be input-to-state stabilized. A controller design approach is proposed in this paper for guarantee of the existence of such switching strategies.