We discuss about the multiobjective double bundle method for nonsmooth multiobjective optimization where objective and constraint functions are presented as a difference of two convex (DC) functions. By utilizing a special technique called the improvement function, we are able to handle several objectives and constraints simultaneously. The method improves every objective at each iteration and the improvement function preserves DC property of the objectives and constraints. Once the improvement function is formed, we can approximate it by using a special cutting plane model capturing the convex and concave behaviour of a DC function. We solve the problem with a modified version of the single-objective double bundle method using the cutting plane model as a objective. The multiobjective double bundle method is proved to be finitely convergent to a weakly Pareto stationary solution under mild assumptions. Moreover, the applicability of the method is considered.