C172 Flight Instructor reported a NMAC during initial climb.

Narrative

My student and I were taking off from runway XX at ZZZ; at approximately XA:09; just after ZZZ tower ceased operations for the day. My student and I made the radio call on CTAF to announce our intentions and depart to the NE; heading to ZZZ1. After we took off; a Cessna Citation announced their intentions to take the runway and I didn't think much of it at the time; but during our climb out they called to ask which way were were turning; I confirmed to the NE; they asked again which way we were turning; and I confirmed we intended to make a left turn to the NE. We were still in our initial climb out on rnwy heading when the citation made their departure call and caught up to us. Flew straight out towards us as I watched traffic alert on our MFD (Multi-function Flight Display) alert us to traffic within 200ft. I took the flight controls and turned the aircraft left; turned my head to look out the window and saw the citation abeam us upon reaching TPA (Traffic Pattern Altitude) and it turned right towards the southwest. Once it turned; it disappeared off the MFD and we could not find any trace of the plane afterwards on flight tracking/filed departures; had to listen to recordings to find and confirm the tail number of the aircraft. All of this happened within the span of a minute; per the recording time stamps.