How did Adam Warlock become The Living Tribunal?

As per Shreedar's comment, this is an alternate universe Adam Warlock who merged with the Adam Warlock of Earth-616. The wiki explains how he became the Living Tribunal in Thanos: The Infinity Finale as follows:

Having accidentally destroyed the entire universe when Pip the Troll awakened him in order to rescue him from Annihilus, a scared Adam Warlock summoned Thanos to his presence in the nothing that was left in his wake to know what he should do next. As he couldn't bring the universe back as it was, Warlock was convinced by Thanos to seek the aid of the Above-All-Others, which was only possible because Warlock had absorbed the essence of the Living Tribunal of his universe, along with everything else, into himself.

Accompanied by the Mad Titan, Warlock had an audition with the Above-All-Others, who was convinced by Thanos to strike a deal with them to allow them to restore the universe as it was before Annihilus' uprising against the Posiverse in exchange for the flavor of an all-new, all-different universe. With the army of the dead by their side, Warlock and Thanos defeated Annihilus' forces and submitted him. Warlock punished Annihilus by devolving him into an early stage of his evolution, which prompted Thanos to kill him by stomping on him, and took away Mephisto's mouth to prevent him from doing any trickery again.

With the universe fully restored and with only the most powerful beings remembering what never happened, Warlock resurrected the original version of himself from that actuality, and proceeded to become the new Living Tribunal as part of the deal he struck with the Above-All-Others.