The assembly is used in leftward- and rightward-retractable door units, and provides wide variations of key to prevent the assembly from being illegally unlocked. In the assembly: a left 25 and a right 28 latch plate are supported on pivots 22 and 54, respectively; a lock casing 17 connected with a thumb turn 15 has locking blades 18, 48; locking projections 30 and 31 abutting against the blade 18 are in the plates 25 and 28, respectively; an interlocking shaft 35 receiving torque transmitted from a front 6 and a rear 9 handle has a driving plate 40; a driving projection 39 abutting against the arms 37, 38 is in the plate 40; a locking hole 50 of a core 20 received in a lock casing 17 assumes a regular hexagonal shape in cross-section, provided that one of its six corners forms a curved surface 50a; a locking pin 51 extends from an inner surface of the hole 50; and, a locking projection 53 is in a center of the bottom of the core 20.