When you mean carbon, I'm assuming you mean inert graphite electrodes. The negative ions will move towards the anode, where oxidation occurs. The positive ions will move towards the cathode, where reduction occurs. From my memory of electrolysis, I think the reaction at the cathode would be Cu2+ + 2e- --> Cu (s). The reaction at the anode should be 2OH- --> 1/2O2 (g)- + H2O + 2e-.