The generalized synchronization （GS） of two different unidirectional coupled Lorenz systems is studied. According to the method of auxiliary-system, by using the theories of stability and the boundary of the responsed system, a sufficient criterion is rigorously proven. Furthermore, based on the modified system approach, GS is classified into three types, the first type,the second type and the third type of GS when the modified system has an asymptotically stable equilibrium of zero solution, asymptotically stable equilibrium of non-zero solution, asymptotically stable limit cycles, respectively. Moreover, using the Routh-Hurwitz theorem to analyze the stability of equilibrium of the modified system, the existence of the first type and the second type of GS are strictly theoretically proved. Numerical simulations show the effectiveness of the method.