Yes, that is correct. In an electrochemical cell, oxidation and reduction reactions occur simultaneously at the anode and cathode, respectively. The species that is oxidized at the anode is the reducing agent, while the species that is reduced at the cathode is the oxidizing agent.
In contrast, in an electrolytic cell, the process is reversed. An external power source is used to drive a non-spontaneous chemical reaction, which allows the electrolytic cell to perform electrolysis. In this case, the species that was oxidized in the electrochemical cell (at the anode) is now reduced at the cathode in the electrolytic cell, and the species that was reduced in the electrochemical cell (at the cathode) is now oxidized at the anode in the electrolytic cell.
So, the species that undergoes oxidation in an electrochemical cell will undergo reduction in an electrolytic cell, and vice versa. The change in the direction of the reactions is due to the change in the driving force of the cell (spontaneous vs. non-spontaneous) provided by the external power source.