DA40 flight instructor reported a door warning indication appeared after takeoff. While the instructor was readjusting the door lock; the door detached from the aircraft. The instructor and trainee returned to the airport.

Narrative

I had a scheduled lesson with my student for XA:00; I supervised him as he went through his pre-flight duties. Upon starting the plane I had him set up the G1000 and call for our IFR clearance. During Taxi and runup all engine indications were in the green and the only PFD annunciation noted was the pitot heat off. On takeoff all engine indications were in the green; I continued to monitor my student's airspeed and engine indications which all looked normal. At 400 feet he raised the flaps; pulled the prop lever back to 2;400 and began his turn to heading 335. Shortly after contacting Departure I took controls so he could put on his view limiting device. I engaged the autopilot and set it to climb at 82kts to an altitude of 6;000 feet and set the heading mode to 335. My student then said the door open indicator had come on; I asked him to take controls back so I could assess the situation. When I turned to look the door latch was perpendicular to the door (open position) and the safety latch was engaged. I attempted to push the door latch to parallel to lock the door closed. Upon closing the latch the door disengaged and became removed from the airplane all within a second of closing the latch. To my best estimation; the door came off somewhere between 2;800 ft and 3;100 ft MSL. At this point I took controls back from my student and [requested priority handling]. We promptly turned back towards the airport and landed on [Runway] XX. No adverse control characteristics were monitored during the flight back; and descent to land. We promptly taxied to our operator's maintenance facility and shutdown the plane.