  We prove that the solutions to the initial-value problem for 2-dimensional
Schr\"odinger maps are unique in $C_tL_x^{\infty} \cap L_t^{\infty}
(\dot{H}^1_x\cap \dot{H}^2_x)$. For the proof, we follow McGahagan's argument
with improving its technical part, combining Yudovich's argument.
