Single engine instructor pilot reported a NMAC while in the traffic pattern. The Tower informed the pilot the other aircraft did not follow instructions.

Narrative

My student (Private Student) and I were practicing in the pattern doing takeoffs and landings. When we were on the beginning of the left downwind for XL. We noticed traffic coming directly at us that had just departed from Runway XR that was doing a left turn approximately on our midfield downwind. Once we noticed it I took controls and immediately closed the throttle and pitched down to lose altitude fast to avoid them colliding with us. Our traffic advisory had gone off as I took controls and started to descend. I contacted Tower and let them know we just had a near-miss on the downwind and asked if traffic was any factor after that. They tried to contact the airplane that almost collided with us; they had switched frequencies already. When they came back into the pattern to do a full stop terminate landing; Tower than advised them to give them a call for possible pilot deviation. My student and I continued to do takeoffs and landings in the pattern and then when we did a full stop terminate to taxi back to the ramp. Tower gave us a number to call when we got back inside. We gave Tower a call they told us that the airplane that almost collided with us did not follow instructions. They were told to make a right turn out on course instead they made a left turn out which then positioned them directly to us. The other pilot said that they had us on their ADSB in; but in my opinion they did not because we barely were able to avoid them. My student was able to handle instructions from me when I said 'my controls' to avoid the collision and was able to be scanning for traffic like he was taught even at a towered airport when the traffic is supposed to be controlled. Tower had stated that they were looking down at the time of the incident and when they looked up the plane was going the wrong direction and was in line with us and the saw us avoid the collision. Tower advised the actions we could take to file the incident and wanted to inform us we did everything right in that scenario and described to us exactly what happened to cause the incident.