Cooperative reconfigurable intelligent surfaces (RISs) are promising technologies for 6G networks to support a great number of users. Compared with the fixed RISs, the properly deployed RISs may improve the communication performance with less communication energy consumption, thereby improving the energy efficiency. In this paper, we consider a cooperative unmanned aerial vehicle-mounted RISs (UAV-RISs)-assisted cellular network, where multiple RISs are carried and enhanced by UAVs to serve multiple ground users (GUs) simultaneously such that achieving the three-dimensional (3D) mobility and opportunistic deployment. Specifically, we formulate an energy-efficient communication problem based on multi-objective optimization framework (EEComm-MOF) to jointly consider the beamforming vector of base station (BS), the location deployment and the discrete phase shifts of UAV-RIS system so as to simultaneously maximize the minimum available rate over all GUs, maximize the total available rate of all GUs, and minimize the total energy consumption of the system, while the transmit power constraint of BS is considered. To comprehensively solve EEComm-MOF which is an NP-hard and non-convex problem with constraints, a non-dominated sorting genetic algorithm-II with a continuous solution processing mechanism, a discrete solution processing mechanism, and a complex solution processing mechanism (INSGA-II-CDC) is proposed. Simulations results demonstrate that the proposed INSGA-II-CDC can solve EEComm-MOF effectively and outperforms other benchmarks under different parameter settings. Moreover, the stability of INSGA-II-CDC and the effectiveness of the improved mechanisms are verified. Finally, the implementability analysis of the algorithm is given.