From what I can see the oil cooler isn't plumbed into the oil system. One hose on the cooler comes up by the frame and ends, goes nowhere.
The other line goes down out of sight by the front of the engine.
If we could see a pic of the right side of the engine showing the engine cover we might be able to see where that other hose goes.
On the right side engine cover it should have been modified so the oil flows out of the engine, a bit more than half way down the front of the cover is were the oil output usually is, to the cooler and back into the engine either through a fitting in the small round .oil filter cover, or as some have done, drill a hole into the top of the case above the round oil filter cover and into the oil filter chamber, then weld in a fitting and tubing into the oil filter chamber.
I chose the fitting in the cover approach. I may try the drill a hole into the chamber and weld in the tube and fitting. It may look a bit more like a stock one might have looked.
Did you just buy this bike? if so the PO may have bypassed the cooler for riding in cooler weather. Don't want the oil too cold.
Leo