Flight Instructor reported an NMAC during initial climb when a Citation jet deviated from their assigned heading and flew toward the reporter's aircraft. The flight instructor took evasive action and after the flight was informed by Tower they did not know why the Citation had deviated off heading.

Narrative

Flew a Solo prep flight with my student. After flying maneuvers in the practice area; we transitioned back to ZZZ to fly some takeoff and landings (TOLs). On our last lap in the pattern; we were cleared for a touch go and right traffic. After takeoff; ATC cleared a Citation for takeoff and told them to 'follow Cessna in the upwind [us].' Before turning crosswind both my student and I visually cleared the area and verbalized it. Just before turning downwind; my student and I noticed a Citation climbing into us; I took controls and expedited a climb. We got a traffic alert from our L3 LYNX ADS-B IN/OUT receiver. After the traffic conflict I asked Tower if that Citation was supposed to be there. Tower then cleared us to turn downwind. I asked for a full stop landing. Tower asked us to call them after the flight. On the phone with tower; they mentioned that the Citation deviated from their assignment of runway heading; and started to descend for us. Tower needed to review the tapes and track log; and would call us back. I was informed that they found no reason why the Citation deviated 20 deg right of assigned heading. Our lateral separation was 307 feet; and had 200 feet of vertical separation.