Low wing aircraft Flight Instructor reported taking evasive action to avoid traffic resulted in a NMAC event.

Narrative

The event happened during my second flight of the day upon arrival back to GYR. We were flying the Gap Arrival at 2;500MSL and I was instructing my SP on the arrival procedures and the associated landmarks. My student made the initial call to GYR tower and received a clearance of left downwind for runway 21. When approximately 1 to 2 miles south of the gap GYR's tower controller informed us of traffic. I looked up and saw what I believed to have been the traffic which was a light sport aircraft north of the gap and above us. My student first noticed the traffic and began turning the aircraft left (north west) and shortly thereafter I heard the G1000's traffic warning and looked up to see an aircraft southbound at approximately our same altitude but slightly offset to the east side of the gap close to the mountains. Taking controls I initially turned left (west) as the traffic as to our right but then shortly after beginning my turn the aircraft turned in our direction southwest bound. This is when I decided to perform an emergency descent and turned away from rising terrain. After descending 500 feet the opposing traffic passed behind us and continued westbound. We continued the gap arrival with no further incident. Additionally; it sounded like the same controller was working both tower and ground frequency at GYR.