Carbon emissions have caused serious environmental pollution, carbon emission reduction (CER) is attracting widespread attention. Many enterprises have not been very effective in CER because of the high cost of CER. Government subsidies and the cooperation of enterprises play an important role in promoting CER. To identify the optimal cooperative emission reduction strategy under cap-and-trade (C&T) and government subsidies, this paper constructs three models to study the behavioural choice of suppliers and manufacturers. First, a static evolutionary game model is constructed between suppliers and manufacturers, and evolutionary stable strategies (ESSs) are discussed. Then, two dynamic mechanisms, dynamic subsidies and dynamic cooperation mechanisms, are constructed and the ESSs of these two models are discussed. Finally, a numerical simulation of the model is conducted to analyse the impact of government subsidies, shared CER cost, CER costs and carbon pricing on the behaviour of suppliers and manufacturers under the three mechanisms. The results show that higher government subsidies can encourage manufacturers to participate in CER activities. Carbon pricing has an incentive effect on the manufacturer's emission reduction behaviour, while has a negative effect on suppliers to cooperate in the three mechanisms. A relatively small cost-sharing ratio is a way to promote upstream and downstream cooperation to reduce emissions. The dynamic cooperation mechanism can be more powerful to promote manufacturers to CER than the dynamic subsidy mechanism. The results provide a theoretical basis for the implementation of CER, governments can formulate reasonable subsidies and C&T policies to enterprises and promote cooperation among supply chain members. Suppliers and manufacturers can implement reasonable decisions to CER and achieve a win-win situation.