Single Engine Pilot reported a NMAC while in the traffic pattern as the aircraft behind overtook them passing 200 ft above necessitating an evasive maneuver to avoid further conflict.

Narrative

At approximately XA:00 am (local) I was conducting a training flight in Aircraft X at ZZZ. We were performing touch and goes in the left traffic pattern for Runway XXL. At mid-field downwind in that pattern I listened to the north Tower instruct a traffic; to follow us in the pattern (we were approximately 1 mile ahead of the traffic at that time). Tower also advised the traffic that they were currently established at a faster ground speed than us; and were converging on us. From that point on; I attempted to visually locate the aircraft behind us; without success. Approximately 1 minute after that; our Traffic Alert Warning System began to enunciate about a traffic conflict with that traffic. We descended to 2200 ft. MSL in an attempt to create vertical separation between us and traffic. I observed on my traffic map that the traffic was still catching up to us; and flying along our direct track.I then observed the traffic fly directly over the top of us; with an estimated separation of about 200 ft. This separation distance was also confirmed by our TAWS system. I immediately advised Tower and turned base; away from that traffic. The Tower contacted the traffic and advised them that they had 'just passed the traffic that they were instructed to follow'. The traffic responded and said that they 'had the traffic in sight the whole time'. With a high wing aircraft versus a low wing aircraft; as was this incident; there is no possibility that either of us had the other in sight when they were directly over the top of us. We were instructed to change to Runway XXR and switch to the south Tower frequency. I advised the south Tower of my intention to file a report for a near miss; and was given a phone number to contact on the ground. After this flight; I contacted the ZZZ Tower phone number at XA:53 am; and spoke to two supervisors.