PROBLEM TO BE SOLVED: To provide a dampening unit for an offset printer for avoiding undesired profile transfer of a print design to a plate cylinder by a form-dampening roller by remarkably improving printing quality by equivalently supplying dampening water to the cylinder without disturbing. SOLUTION: An adjacent dampening water supply unit 12 is disposed at the front side as a first contact position at a contact position of at last one form-dampening roller 5 adjacent to a plate cylinder 1 with the cylinder 1, and a first roller 9 having a non-adhesive coating layer for repelling dampening water and ink is disposed at the rear side as a second contact position of the roller 5 so that at least the roller 9 is rotated equal or different circumferential speed to or from the circumferential speed of the roller 5 while oscillating a least the roller 9 in an axial direction.