In this paper, the output synchronization in large-scale discrete-time networks is examined by utilizing the novel phase tool, where the agent dynamics are allowed to be significantly heterogeneous. The synchronization synthesis problem is formulated and thoroughly investigated, with the goal of characterizing the allowable heterogeneity among the agents to ensure synchronization under a uniform controller. The solvability condition is provided in terms of the phases of the residue matrices of the agents at the persistent modes. When the condition is satisfied, a design procedure is given, producing a low-gain synchronizing controller. Numerical examples are given to illustrate the results.