Surrogate-assisted evolutionary algorithms have received a surge of attentions for their promising ability of solving expensive optimization problems. Existing surrogate-assisted evolutionary algorithms usually adopt the regression models and the binary classification models to guide the evolution of the population for solving the multiobjective optimization problems. However, the regression models will make the algorithm to be increasingly computation-expensive as the number of objectives increases, while the use of the binary classification models might suffer from the poor diversity since the diversified information of solutions cannot be reflected in these classification models. For this issue, this paper proposes a surrogate-assisted many-objective evolutionary algorithm using the cooperation of the multi-classification and regression models to improve the search quality while reducing the computational cost. Our approach includes two parts: At the model training stage, a multi-classification model is constructed to divide the whole population into several classes for ensuring diversity, a distance regression model and an angle regression model are used to select solutions with better convergence and diversity in each class; At the evolution stage, a coevolutionary framework is used to guide the evolution according to a new selection criterion. Experimental results verify the effectiveness of the proposed algorithm on a set of expensive test problems with up to 10 objectives.