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Magnetic force is caused by the movement of charge. However, it is more fully described as a manifestation of a single electromagnetic field. This field can appear as an elctrostatic field or a magnetic field depending on one's frame of reference. The properties are described by Maxwell's equations and these equations are invariant under a Lorentz transformation and so are consistent with special relativity. I'm not sure whether a simpler or a more thorough explanation is required.