C172 flight instructor reported an NMAC while maneuvering in the traffic pattern of a non-towered airport.

Narrative

We had just taken off Runway XX at ZZZ and was climbing up to our left crosswind altitude when I heard an airplane on the 45 entry for left downwind Runway XX and an airplane crossing over mid field for a teardrop entry into left downwind Runway XX. We made our radio call and turned left crosswind and had the first 45 entry traffic insight which was no factor. We then heard the second airplane that was doing a teardrop entry say they were gonna hold off on their entry because of traffic; so I turned my attention back to my student who had climbed up to 1;850 [feet] instead of 1;700 [feet] and told him we needed to get back to 1;700 [feet]. We made a radio call for left downwind Runway XX and made our turn while still scanning for the traffic that said they were not going to do their teardrop entry into the left downwind; we didn't see any traffic partly because the sun was in our eyes; so I glanced down at my iPad and saw the traffic was directly 200 feet below us. At that exact moment; they came on the radio and said they were directly below us; I guess they saw us on their ADS-B as well and said they were breaking out of the pattern to the right. At that point; I had gotten a visual on their aircraft and as they made a right hand turn away from us they became no factor. With all the factors of a busy cockpit; training flight; several planes in the traffic pattern; the sun directly in our eyes when we turned crosswind we probably should have extended out on our climb and had a visual on the teardrop entry traffic; even if they said they were not going to execute their entry; before we turned left crosswind and put the sun right in our face.