Goh's Eternatus was caught by
Goh with the help of
Ash Ketchum as well as Zacian and Zamazenta. It was was temporarily stored deep underground in the basement of the institute by
Professor Magnolia so Galar would be kept safe until they could figure out what to do with it. Eternatus' energy flows through the Galar region and it is deeply entwined to the Galar particle. It's a Pokémon who's inseparable from the lives of the people of Galar and the people must coexist with it.
On the day before the Masters Tournament finals between
Ash Ketchum and
Leon,
Professor Magnolia met with
Goh and it was revealed that
Leon had been helping Eternatus get used to people. It was tough in the beginning with all the rampaging, but they finally were able to calm it. After realizing the importance of the Pokémon to the Galar Region,
Goh could see that Eternatus would approve of
Leon as its Trainer but he vowed to work hard to become a great Trainer himself to be worthy of its approval. It eventually was entrusted to
Leon. (See
Leon's Eternatus).