GA pilot in the pattern at 7S9 reported an NMAC with a light aircraft operating at UAO.

Narrative

I was turning left downwind for rwy 02 at 7s9 after a go-around due to gusts. I was climbing through 800' to TPA (1000'). Just after turning downwind I got an audible ADSB traffic alert indicating 'very close' at my 2 o'clock. I saw on my display the traffic to be to my right/west and going behind me at the same altitude. I never visually saw the traffic. No radio transmissions were made by the other aircraft; Aircraft Y; on 7s9 CTAF. I reviewed the ADSB recording and saw that Aircraft Y was on a right downwind for rwy 35 at UAO. Aircraft Y conducted a left 360 which placed it below and outside the UAO class c shelf. This left 360 put Aircraft Y less than a mile from 7s9; going the opposite direction of downwind traffic at 7s9; and at near the same altitude as 7s9 traffic at TPA. As I was not monitoring UAO XXX.XX while landing; it is unknown but presumed that UAO tower directed Aircraft Y to make the left 360 into the 7s9 traffic pattern.The horizontal distance between me and Aircraft Y was about 500'.