This paper presents the design and synthesis of star-shaped copolymers with two poly[2-(diethylamino)ethyl methacrylate] and poly(ethylene glycol) blocks linked via a disulfide bond and each end of the four arms capped by folic acid (FA), from which the prepared nanodrug carriers simultaneously possess pH/redox dual response and active targeting functions. The polymer micelles exhibit excellent stability as reflected by their low critical micelle concentration values of 1.03-2.51 mg/L. The doxorubicin (DOX)-loaded polymer micelles are in the range of 108 to 143 nm, and the DOX-loading capacities are found to be up to 32.3%. The sensitive pH and redox responses are demonstrated by examining the drug release behaviors under the varied acidic condition from pH 7.4 to 5.0 and the glutathione concentrations from 0 to 10 mM, respectively. Moreover, the observation of confocal laser scanning microscopy confirms that the functionalization of arm ends by FA indeed enhances the internalization of DOX-loaded micelle particles in HeLa cells. As a result, the DOX-loaded nanocarriers can deliver therapeutic drugs to target HeLa cells, and the viability of HeLa cells (10.1%) approaches the value of the pristine DOX (9.98%), showing promising application as drug delivery nanocarriers for safe and highly efficient cancer therapy.