PA28 Flight Instructor on short final to a non-towered airport reported a NMAC with another aircraft on short final flying 100 feet below them.

Narrative

This event occurred during a training flight with my student. I am a CFII and my student is currently working on an instrument rating. We were completing an approach RNAV RWY XX into an untowered field; ZZZ. We received approach clearance from ATC; switched comms to CTAF; and made a 3.0 nm final call on CTAF. We did not hear any other traffic on CTAF nor did we see traffic displayed on our MFD (Multi-Function Flight Display). I was the pilot flying. I wanted to fly an approach so my student could observe and practice teaching me how to make corrections during my approach. My student has been progressing well through the instrument rating and I wanted to give him a different learning experience by having him practice being an instructor and walking me through how to fly an approach. During final; approximately 1.5-2.0nm from the threshold we receive a traffic alert. I look at the MFD and the traffic was showing 100 feet below us. At first; I thought it was our on shadow as I was making a lot of pitch corrections to show my student how much the advisory glidepath deviates and becomes more sensitive the closer we get to the runway.I asked my student if he saw any traffic. My student said he could not. Nor could I. I took a few seconds to make a decision; but I decided to be safer and go-around and not assume it was our shadow. Once we initiated the go-around and were climbing on the upwind; we hear a Cessna make a CTAF call that they were clear of RWY XX. We continued the go-around; joined a left traffic pattern (making crosswind; downwind; and base calls on CTAF) and landed without incident. On our downwind; we heard the Cessna say they were taking off and making left traffic. During our left base call; the Cessna made an incorrect call that they were also on base before correcting themselves and saying they were on left crosswind. Contributing to the problem was a lack of the Cessna making CTAF calls during their pattern work and the fact they were a high-wing and we were a low-wing aircraft. I carry a SENTRY with me but on instrument approaches; I leave my Foreflight view on the approach plate. In the future; I will make sure to use the MAP function in order to see traffic the entire flight including during short final. I will also never hesitate again to initiate a go-around when in doubt.