A rotary-tool mandrel for a unit for converting a flat substrate, on which a sleeve (13) is intended to be fitted, the mandrel includes a cylindrical core (14), a peripheral wall (17) that is able to take up a rest position and a locking position by exerting a radial pressure on the sleeve (13) in order to lock it in position on the mandrel (12), a pressure fluid circuit (21) provided between the peripheral wall (17) and the cylindrical core (14) for exerting the radial pressure on the sleeve (13), and a cooling fluid circuit (24) for allowing a fluid to flow in the region of the cylindrical core (14) and for cooling the mandrel (12).