Provided is a door opening and closing system capable of returning to a return state where a catch and a retractable arm are engaged with each other even if the retractable arm is closed completely as a result of incorrect usage such as a prank. A catch has a catch base which is mounted on either one of the door and a frame and a catch shaft which is installed in the catch base slidable in an axial direction and engaged with a retractable arm. At a tip end of the catch shaft, a small-diameter part is provided as a step part. The step part of the catch shaft and the retractable arm are engaged with each other in a state where the retractable arm is further rotated in the closing direction after a main body part of the catch shaft and the retractable arm are engaged with each other.