The introduction of the carbon market has changed the power generation cost of traditional fossil fuel units, thereby affecting bidding behavior in the electricity market. At the same time, different market players are involved in the transmission and distribution network, and it is necessary to study the market trading method of that network in the carbon-electricity market. This paper proposes a carbon-electricity coupling market trading decision-making method based on transmission and distribution coordination. First, a trading framework for the carbon-electricity coupling market considering the coordinated operation of transmission and distribution networks is proposed. Secondly, a two-layer trading model of the carbon-electricity coupling market based on multi-agent transaction based on transmission and distribution coordination is constructed. The upper models are the carbon-electricity collaborative trading decision-making models of each distribution market and power generators, and the lower models are the clearing models of the transmission and carbon markets. This quantifies the relationship between carbon quota and power generation. Then, the Karush-Kuhn- Tucker (KKT) condition, the duality principle and the Big-M method are used to transform the bi-level model of each trading agent into a single-layer mathematical model with equilibrium constraints (MPEC), and all MPECs constitute an equilibrium problem with equilibrium constraints (EPEC) model, which is analyzed by the diagonalization method. Finally, case analysis is carried out based on the T6-D7-D9 and T30-D33 systems. The results show that the proposed method can effectively promote the reduction of carbon emissions, reduce the total operating cost and improve social welfare in the power industry.