In this technical note, we study the cooperative global robust output regulation problem for the unity relative degree nonlinear multi-agent system in normal form with a nonlinear leader system. The same problem was studied recently under the assumptions that the system contains only polynomial nonlinearities, the leader system is linear and the communication network is undirected. By employing a distributed nonlinear internal model, we will construct a control law that accommodates non-polynomial nonlinearities, nonlinear leader systems, and directed communication networks. Our result will be illustrated by an example with the well known Van der Pol oscillator as the leader system.