We investigate the principles of  fully autonomous and quantum refrigerators that are powered by a quantum device. They may operate in a dual mode: as  ``Carnot'' or ``absorption'' refrigerators. The operation mode is determined by the initial quantum state of the piston/battery. Its coefficient of performance may  surpass the classical bound, while adhering to the second law. These general results are illustrated for the minimal quantized model of a refrigerator comprised  of a qubit coupled to a harmonic-oscillator piston and two spectrally separated baths.