Neuromorphic vision that integrates the functionalities of sensing, memory, and processing may provide an important approach to overcome the drawbacks of a conventional artificial visual system such as data redundancy and high-power consumption. It has exhibited considerable potential to mimic the functionalities of human vision even beyond the visible-light range. In this work, we show an ultraviolet (UV)-responsive synaptic transistor based on the heterostructure of 4H-SiC and organic semiconductors. Benefiting from the heterostructure design and photogating effect, the nonvolatility of the synaptic transistor is achieved. Various biological synaptic functionalities are successfully mimicked by the synaptic transistor. The electrical energy consumption of the device per synaptic event is only 0.55 fJ, which is comparable to the energy consumption of a biological synapse. Furthermore, the dynamic learning and forgetting process of the image of a letter is simulated by using an array of the devices, demonstrating the potential of the UV-responsive synaptic devices for neuromorphic UV vision with the capability of the image recognition and memory.