The Pilgrimage of a Summoner is the name of the journey that a Summoner makes in order to defeat Sin.

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The Summoner makes a journey and stops at numerous temples with their Guardians. At these temples, the Summoner prays to the fayth of the temple and recieves an Aeon. After being granted assistence by the Aeons aquired from the temples, the Summoner makes their way to the temple in the Zanarkand Ruins. Upon completing the trials of this temple, they are required to choose one of their Guardians to become the Final Aeon, who is then used in the actual battle agains Sin, a battle in which the summoner ultimately dies, regardless of the battles outcome. The time period after this event is called the Calm, and is a time where Sin is no more. After this Calm, however, the Final Aeon becomes the new Sin and starts the cycle anew.

Yuna became the first Summoner to break the final process in the Pilgrimage. Yuna did not receive her Final Aeon, as she was not willing to sacrifice one of her loved ones. She instead traveled inside Sin in order to defeat the source, Yu Yevon, the creator of Sin, thus ending the need for the Pilgrimage and bringing an Eternal Calm, which could not have been achieved, had she received the Final Aeon.

Spoilers end here.
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