General Aviation Flight Instructor with a Student Pilot reported an NMAC during approach to a non-towered airport; FCH; requiring evasive action.

Narrative

I was on a training flight with one of my students. That day the mission was takeoffs and landings. My student is a couple flights away from solo; so at the time he's the one in control of the aircraft. He announces the left downwind leg right before initiating the turn as we are on the left crosswind leg. Right as we go wings level on the downwind leg; the other aircraft made initial call to the airport reporting a 2 mile 45 degree entry into the left downwind for runway 30. At no point did he advise whether he had me in sight or not. After hearing what he announced I immediately looked off to the right to look for him and he passes right underneath us; at that point I immediately took controls of the aircraft and my student verified I had the controls. My quick reaction was climbing and we gained 300 ft. but that point the aircraft has already passed us. I asked if he'd seen us and his response was that he did and he's going to be leaving the traffic pattern and rejoining it. That was it. The aircraft wasn't in 45 degree to the airport; they were more so at a 90 degree. Me and my student both agreed that the lesson is over. Upon the completion of the landing and securing the airplane. We tried to approach him and discuss what has happened he was very dodging of talking about it and was offering me to teach in his aircraft. We ended up leaving him alone and not discuss it any further.