This paper presents an efficient evaluation algorithm for cross-validating the two-stage approach of KFD classifiers. The proposed algorithm is of the same complexity level as the existing indirect efficient cross-validation methods but it is more reliable since it is direct and constitutes exact crossvalidation for the KFD classifier formulation. Simulations demonstrate that the proposed algorithm is almost as fast as the existing fast indirect evaluation algorithm and the twostage cross-validation selects better models on most of the thirteen benchmark data sets.