The Bourne Identity

In the Mediterranean Sea, Italian fishermen rescue an unconscious American (Matt Damon) floating adrift with two gunshot wounds in his back. They tend to his wounds, and when the man wakes, they find he suffers from dissociative amnesia. He has no memory of his own identity, while he retains his speech and finds himself capable of advanced combat skills and fluency in several languages. The skipper finds a tiny laser projector under the man's hip that, when activated, gives a number of a safe deposit box in ZÃ¼rich. Upon landing, the man heads to investigate the box. Arriving at the bank, the man finds the box contains a large sum of money in various currencies, numerous passports and identity cards with his picture on all of them, and a handgun. The man takes everything but the gun, and leaves, opting to use the name on the American passport, Jason Bourne.
After Bourne's departure, a bank employee contacts Operation Treadstone, a CIA black ops program. Treadstone's head, Alexander Conklin (Chris Cooper), activates three agents to take down Bourne: Castel (Nicky Naude), Manheim (Russell Levy), and the Professor (Clive Owen), while also issuing alerts to local police to capture Bourne. Meanwhile, CIA Deputy Director Ward Abbott (Brian Cox) contacts Conklin about a failed assassination attempt against exiled African dictator Nykwana Wombosi (Adewale Akinnuoye-Agbaje). Conklin promises Abbott that he will deal with the Treadstone agent who failed.
Bourne attempts to evade the Swiss police by using his U.S. passport to enter the American consulate, but he is discovered by guards. He evades capture, leaves the embassy, and gives a German woman, Marie Kreutz (Franka Potente), $20,000 to drive him to an address in Paris listed on his French driving license. At the address, an apartment, he hits redial on the phone and reaches a hotel. He inquires about the names on his passports there, learning that a "John Michael Kane" had been registered but died two weeks prior in a car crash. Castel ambushes Bourne and Marie in the apartment, but Bourne gets the upper hand. Instead of allowing himself to be interrogated, Castel throws himself out a window to his death. Marie finds wanted posters of Bourne and herself, and after agonizing, agrees to continue to help Bourne. After a chase in which Bourne evades Paris police in Marie's car, the two fugitives spend the night together in a Paris hotel.
Meanwhile, Wombosi continues to obsess over the attempt on his life. Conklin, having anticipated this, planted a body in the Paris morgue to appear as the assailant, but Wombosi is not fooled and threatens to report this. The Professor assassinates Wombosi on Conklin's orders. Bourne, posing as Kane, learns about Wombosi's yacht, and that the assailant had been shot twice in the back during the escape, just as he was. Bourne now considers himself to have been the assailant. He and Marie take refuge at the French countryside home of her step-brother Eamon (Tim Dutton) and his children. Under pressure from Abbott to tie off the Wombosi matter entirely, Conklin tracks Bourne's location and sends the Professor there, but Bourne shoots him twice with Eamon's shotgun, mortally wounding him. The Professor reveals their shared connection to Treadstone before dying. Bourne sends Marie, Eamon, and Eamon's children away for their protection, and then contacts Conklin via the Professor's phone to arrange a meeting. From a rooftop near the arranged location in Paris, Bourne sees Conklin has brought backup, so he abandons the meeting, but uses the opportunity to place a tracking device on Conklin's car, leading Bourne to Treadstone's safe house.
Bourne breaks in and holds Conklin and logistics technician Nicolette "Nicky" Parsons (Julia Stiles) at gunpoint. When Conklin begins pressing him to remember his past, Bourne recalls his attempt to assassinate Wombosi through successive flashbacks. As Kane, and working under orders from Treadstone, Bourne infiltrated Wombosi's yacht but could not bring himself to kill Wombosi while Wombosi's children were present, and instead fled, being shot during his escape. Bourne announces he is resigning from Treadstone and is not to be followed. As agents descend on the safe house, Bourne fights his way free. When Conklin leaves the safe house, he encounters Manheim and is killed as the latter is operating under Abbott's orders. Abbott shuts down Treadstone.
Abbott reports to an oversight committee that Treadstone is "all but decommissioned" before discussion turns to a new project codenamed "Blackbriar". In the final scene, Bourne finds Marie renting out scooters to tourists on Mykonos, and the two reunite.

Question: What information does the Professor share with Bourne as he's dying?
Answer: their Treadstone connection