In fact during the story, when Taki switch with Mitsuha, Taki go in the past. So the present's Taki is 3 years in the future and in his present Mitsuha is dead. I've right ?
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Taki lives in 2016. Mitsuha lived in 2013, before she died. That changed when Taki drank the sake that contained half of her soul. Suddenly, she revived in 2013, causing a butterfly effect. Taki is 3 years younger than Mitsuha normally. But when they met at the volcanic crater during twilight, they were both 17. According to the movie, multiple worlds can collide during twilight.
In the first timeline, Taki and Mitsuha switch bodies, being seperated by three years in time - Mitsuha in 2013, and Taki in 2016. In that timeline, Mitsuha is killed by the comet fragment, but Taki is able to remember.
At Kateware-doki or twilight, both are able to meet each other, it seems, in a place where time has either stopped or simply does not exist, until the real-time ends of the period are reached.
After which, both are transported back to their own timelines, and Taki quickly begins to forget because the history of Itomori's residents are quickly being re-written. So does Mitsuha, who attempts to resist the flow of time through mental strength of remembering Taki. She fails, but is able to give herself enough drive to confront her father and save the town.
However, there is a mild paradox that happens here due to Musubi - Taki and Mitsuha are able to remember the sight of one another because of the logic of Musubi - that it controls time - that time weaves, and stretches, and reconverges again. Therefore, the ending is logical enough to happen.