Pilot reported an airborne conflict with an aircraft that was departing a crossing runway.

Narrative

My student and I were in the pattern at Komomo Municipal Airport. After our initial takeoff and climb out I was instructing my student on patterns and proper procedures. I had heard on the radio two aircraft in the area. Aircraft Y calling a 10-mile final for Runway 05 and I heard Aircraft Z make a taking off Runway 14 call. The winds at the time were favoring Runway 05. My student and I had already made a previous takeoff and landing on Runway 05 for pattern practice. After I had heard the plane taking off Runway 14 I actively tried to search for him and could not spot him so I started to divert my attention to the other aircraft landing on Runway 05. My student starts to turn base when the Aircraft Y calls a 5-mile final behind us. We continue down to the runway and make a safe landing. We come to a full stop on the runway to clean up the airplane and make a very short debrief because the Aircraft Y landing behind us was on a 1-mile final as he made a radio call. We make a radio call that we are departing Runway 05 and staying in the pattern. At this point; I am making sure that my student is doing the proper procedure on takeoff. We were approximately 10 kts. below our rotation speed when I look up and to my left and notice that there is an aircraft taking off Runway 14 and we are on a direct collision course if no action was taken by either pilot. I immediately abort the takeoff by quickly pulling the throttle to idle and applying full brakes and taking the controls from my student. We start coming to a stop when I announce on the radio that we aborting the takeoff because of the Aircraft Z departing [Runway] 14. We come to a stop on the runway and watch the Aircraft Z continue their takeoff and continue to not make any radio calls. Aircraft Y I believe was on the ground and makes an 'on the go' call and departing the area to the south. Aircraft Z departs the area and we continue our lesson with no further incident. Later after the flight; the pilot of the Aircraft Y contacted me on my cell and called to fill each other in. I could tell that my aborting the takeoff and his subsequent go-around really shook him up by the sound of his voice. We talked on the phone about each other's perspectives and he said that because of the departing Aircraft Z and my aborted takeoff on his go-around he almost stalled the airplane and crashed it behind me or into our aircraft. He was going to stay in the pattern behind us but decided to go back to his airport after the event. This event I believe may have been prevented in a couple of ways. The Aircraft Z pilot should have been listening to the radio and should have heard that there was a plane already in the pattern and a plane landing on Runway 05. They should not have taken off of the runway they did. They could have waited until both aircraft were safely out of the way and back into the air before departing the runway. There is a chance that during my instruction with my student that I may have missed the radio call by the Aircraft Z. I maybe should have stopped talking and listened. The Aircraft Y also could have given me a bit more time to assess my situation and give my student and me some more time to take back off.