Flight Crew Commercial; Flight Crew Flight Instructor; Flight Crew Instrument; Other CFII
Flight Crew Last 90 Days 12; Flight Crew Total 730; Flight Crew Type 550
Situational Awareness; Training / Qualification; Confusion; Communication Breakdown; Troubleshooting; Workload
Conflict NMAC; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy
Air Traffic Control Separated Traffic; Flight Crew Requested ATC Assistance / Clarification; Flight Crew Took Evasive Action
Human Factors
We were coming back to ZZZ (our base) from ZZZ1. We were assigned ZZZZZ Arrival. Shortly after crossing the shoreline; Tower told us to maneuver toward a landmark to follow the traffic. After we made our turn; Tower asked us to if we have the light transport traffic in sight; my student pointed out a traffic to me and I radioed the tower that I had the traffic in sight. Tower then told us to cross behind the light transport traffic and cleared us for Runway XXL. After the traffic my student pointed out passed in front of us we angled toward the base leg. then I realized that traffic is not a light transport. Then I inspected the final for Runway XXR and saw a light transport about 250 ft. below; and about 500 ft. north east of us. I took over the airplane and maneuvered to avoid flying in front of the light transport. [The] maneuver was successful and we flew behind them. Shortly after the evasive maneuver Tower asked us again if we have the light transport traffic in sight; I told them 'We do have [light transport] in sight' We proceeded to land on Runway XXL.In retrospect; I shouldn't have told the tower we have the traffic in sight without positively identifying the light transport. A good radio transmission  would be 'we do have a traffic over X location please confirm that is our traffic.'
Flight Instructor reported a NMAC in the traffic pattern. The Instructor and student misidentified the aircraft they were to follow per ATC. The Instructor took control and maneuvered the aircraft into the correct traffic pattern flow.