Flight Instructor with student on downwind at an airport where the Tower was closed reported a NMAC when a subsequent departing aircraft turned crosswind towards them.

Narrative

In the traffic pattern for RWY 35 in Aircraft X. UAO tower closes at XA:45 on Fridays so the airport was uncontrolled at the time. My student was flying and making all appropriate radio calls. We had just taken off after our 2nd landing and radio calls had been made to indicate we were taking off to remain in the patterns and had turned crosswind. Prior to our downwind call; I identified same altitude traffic flying in the same direction as the downwind; but it was too wide for me to know the intentions for sure. I was helping my student to identify that traffic when another aircraft suddenly crossed right in front of our flight path; coming low and from the left. It was incredibly close and fast we had no time to react. Our aircraft does not have ADSB in but I have a Sentry attached to my iPad so I was able to identify the plane as Aircraft Y. This traffic had been holding short of the runway during our last landing and had taken off after us. Due to his higher performance; he turned crosswind shortly after passing the numbers for 17. His crosswind intersected our downwind path. I do not recall hearing a single radio call from him. I made a radio call asking the same altitude traffic further out his intentions and informed Aircraft Y how he had almost collided with us and he said something about how he was just turning his crosswind. The further out traffic gave us right of way so we landed and returned back to parking. Aircraft Y flew an odd pattern; did a 360 on final before leaving the area just to turn back in to land.