Izayoi is Inuyasha's mother, though the manga never actually names her. We know surprisingly little about her since she died before the series began, and Inuyasha doesn't talk about her much—probably because her death still weighs on him.
She was a feudal lord's daughter who fell in love with Inu no Taishou, and he loved her back. Whether they were officially married or mated is unclear, but they had a serious relationship. When Inu no Taishou died, she was left to raise Inuyasha alone. She worried constantly about how he'd be treated by both humans and demons because of his mixed heritage. The exact timing and cause of her death remain unknown, but it happened when Inuyasha was still young.
The third movie gives her a proper name that the manga never provided. Sesshomaru, Inuyasha's half-brother, holds a grudge against her memory—he's convinced she weakened their father and made him vulnerable, ultimately leading to his death.
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