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The (2+1)-dimensional two-component nonlocal solitons of coupled nonlinear Schrödinger equation with the partially nonlocal nonlinearity are analytically obtained. Two component optical fields exhibit different types of dynamical behaviors. The dark soliton and bright soliton for two field components describe two co-propagating fields at different frequencies and they are both quickly stabilized to propagate parallel to the z-axis after the initial oscillations. Superposed rogue waves in bright soliton and dark soliton are also constructed for two field components.