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This paper provides synchronization conditions for networks of nonlinear systems, where each component of the network itself consists of subsystems represented as operators in the extended L2 space. The synchronization conditions are provided by combining the input-output properties of the subsystems with information about the structure of network. The work is motivated by cellular networks where signaling occurs both internally, through interactions of species, and externally, through intercellular signaling. The companion paper explores results for state-space models as well as biochemical applications.