Flight Instructor with student reported a NMAC at PUJ non-towered airport requiring evasive action to avoid a possible collision.

Narrative

My student and I were practicing landings at PUJ; we were flying Aircraft X. The weather was clear and the winds were mostly calm. Runway 31 was in use at the time with 3 (possibly 4) planes in the pattern (including us). After a touch and go on Runway 31 we climbed and flew a standard left traffic pattern; planes entering the pattern were doing so by crossing midfield between 2800 and 3000 MSL and making a right teardrop to enter the left down wind for 31.The other plane involved was Aircraft Y that was flying from the northeast; I believe he made a 10 mile call and stated his intentions to overfly the field; at the time his altitude was showing on our traffic page at 2800 MSL; 500 feet above the TPA of 2300 MSL. We made our calls for our turn onto the left crosswind and then for left base. Aircraft Z took off Runway 31 behind us and followed us in left closed traffic. Myself and Aircraft Z behind us were both in the left down wind for 31 at the time of the incident.As we were nearing midfield left downwind for 31; Aircraft Y over flew midfield; our traffic page showed him same altitude and when I got visual contact I; realized he was flying directly at us same altitude. I immediately took controls from my student; set power to idle; and forced the nose of our plane down in order to avoid collision; I also notified the pilot on CTAF that he was at TPA; the pilot acknowledged but did nothing to prevent flying into us. Aircraft Y turned into the left crosswind exactly where our plane was; had these evasive actions not been taken there very possibly would have been a collision. Estimated distance between our two aircraft was 25-50 feet laterally and 200 ft vertically. This pilot behaved recklessly and posed a serious hazard to the safety of my flight.I maintained approximately 1900-2000 MSL as I side stepped to the right of the downwind and maintained visual separation with the aircraft. Once he turned left base we were able to climb and depart the area to the east-northeast.