C172 Flight Instructor reported an NMAC event during landing pattern training. Flight Instructor took evasive action during final landing to avoid the opposite direction aircraft who had mistakenly lined up for opposite runway.

Narrative

I was instructing a student in Aircraft X at ZZZ. We were holding short Runway XX while listening to coms. Two aircraft were approaching the airport and making position calls to enter the pattern for XX. One was an Aircraft Y; the other a local flight school aircraft. The flight school said they would give way since the Aircraft Y was much faster. Aircraft Y stated yes; they were flying 140 kts. Based on their position call; we departed XX to remain in the pattern. Aircraft Y announced entry to the pattern; called base for XX; called final for XX.On climb out; at approximately 1800 ft. MSL and about time to turn crosswind; I glanced to my iPad and noticed an aircraft heading towards us for Runway X. I took controls from my student; put the nose down and made an evasive bank to the right to avoid Aircraft Y. I also made a call on the coms that Aircraft Y was coming at us for Runway X; not XX. Aircraft Y broke off as well and proceeded to fly a right downwind; base to then land on XX. The traffic pattern for ZZZ is left-hand and there were two aircraft flying left pattern at the time.I called the company the plane was registered under. They are a broker and told me they had sold the plane and the owner had recurrent training. The pilot clearly thought he was flying to land on Runway XX while actually descending to Runway X; which is especially concerning since his last indicated airspeed on approach was 129 kts. while I was approaching at 75 kts. Most disturbing is this pilot came back to ZZZ several days later and I spoke with him; indicating I was his near midair the other day. He made light of it and lied; stating he was trying to look for the windsock. If that were the case; he would not have been calling out his pattern legs for Runway XX if he was looking for the windsock; especially since there were two of us in the pattern flying XX already. A pilot this non-challant about the near midair; flying a fast aircraft is a safety hazard. We were lucky that day I had traffic show on my iPad as our climb out puts in a compromised nose high position to see and avoid an aircraft coming right at us; especially when not expecting it based on the current traffic patterns and comm calls by the pilot.