This paper provides synchronization conditions for networks of nonlinear systems. The components of the network (referred to as "compartments" in this paper) are made up of an identical interconnection of subsystems, each represented as an operator in an extended space and referred to as a "species." The compartments are, in turn, coupled through a diffusion-like term among the respective species. The synchronization conditions are provided by combining the input-output properties of the subsystems with information about the structure of the network. The paper also explores results for state-space models, as well as biochemical applications. The work is motivated by cellular networks where signaling occurs both internally, through interactions of species, and externally, through intercellular signaling. The theory is illustrated by providing synchronization conditions for networks of Goodwin oscillators.