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A four-level double-ladder cold atoms system with spontaneously generated coherence trapped in a moving optical lattice is explored to achieve optical nonreciprocity.When spontaneously generated coherence (SGC) is present,the remarkable contrast optical nonreciprocity of light transmission and reflection can be generated at each induced photonic bandgap in the optical lattice with a velocity of a few m/s.However,when the SGC effect is absent,the optical nonreciprocity becomes weak or even vanishing due to the strong absorption.It is found that the optical nonreciprocity is related to the asymmetric Doppler effect in transmission and reflection,meanwhile the degree and position of optical nonreciprocity can be tuned by the SGC effect and the Rabi frequency of the trigger field.