In this paper, we study cooperative robust output regulation problems for a directed dynamical network of diffusively interconnected Lur'e systems that consist of a nominal linear dynamics with an unknown static nonlinearity around it. In the present paper we assume that the nominal linear part of each agent is identical, but the nonlinearities are allowed to be different for distinct agents. In this sense the network is heterogeneous. As is common in the context of Lur'e systems, these unknown nonlinearities are assumed to be passive or sector bounded. The interconnection graph among these agents is assumed to contain a directed spanning tree. Similar to classical output regulation problems, there is an exosystem generating a common reference signal that these agents are required to track cooperatively. Our designed distributed dynamic state feedback protocol makes a copy of the reference signal at each agent asymptotically, and thus the cooperative robust output regulation problem becomes a robust tracking problem that can be handled locally. Sufficient synchronization conditions are given for both the passive unknown nonlinearity case and the sector bounded unknown nonlinearity case, respectively. It turns out that our cooperative protocol is naturally fully distributed.