Yui is the mother of Shinji Ikari and the wife of Gendo Ikari, was a brilliant bioengineer who worked for Project E at Gehirn. Additionally, she is the daughter of a prominent member of Seele.
Yui Ikari is one of the more powerful presences in the series, Yui is shown to be kind, optimistic, and motherly. Her faith in humanity and belief that it is worth preserving and remembering stand out when compared to the more cynical characters on the rest of the cast.
Yui became romantically involved with a mysterious man, several years older than herself, named Gendo Rokubungi, who was constantly the source of shady rumors among them that Gendo had approached Yui for nothing but her talent and her affiliation with Seele. Regardless, in Yui's own words, he was "quite a sweet person. It's just that no one knows it."
Yui took up work at the Artificial Evolution Laboratory in Hakone, Japan, which was really a front for Seele's secret research organization, Gehirn. Her skills in bioengineering went towards the "Adam Revival Project", more commonly called "Project E". In 2004, Yui volunteered herself as the test subject for the Contact Experiment with the incomplete Eva-01 (a decision she had made by autumn of the previous year). She brought the 4-year-old Shinji to the site of the experiment so that she could "show [him] the bright future". The experiment, ostensibly, was part of Gehirn's research into the creation of artificial souls, but it resulted in a "freak accident" instead, wherein her soul was absorbed into the Eva's core and her body reverted to LCL. Gendo subsequently undertook a "Salvage Operation" that attempted to recreate Yui's body and fix her soul back into it, but this failed, instead resulting in the creation of the first Rei Ayanami.
Yui's soul remained inside of Eva-01 throughout the series, with her son Shinji piloting it to defeat the Angels. Despite Nerv's extensive system of restraints, Yui's soul could take control of the Eva's body in situations where Shinji was in extreme danger. Her control manifested itself in berserk mode, a state of greatly increased power and savagery.
She first moved all by herself in Episode 01, protecting Shinji from falling lights and in the following battle in Episode 02, where she took control and defeated Sachiel when Eva-01 shut down after sustaining severe damage from the Angel. In Episode 16, she made contact with her son and took control of the Eva immediately before all power to the life support systems failed, breaking out of and killing the twelfth Angel, Leliel. Finally, she manifested in Episode 19 against the fourteenth Angel, Zeruel. This time, in addition to taking control over the Evangelion and quickly defeating the Angel, she also ingested its S² Engine Engine in the process.
When Third Impact arrived and Rei III rejected Gendo, control of Third Impact was given to Shinji and, ultimately, Yui. After Shinji's rejection of The Human Instrumentality Project, Yui took the Spear of Longinus and floated out towards space, becoming an eternal testament to humanity's existence, as Evangelion Unit-01.
Yui reveals to Gendo that she believes anyone can find Heaven; they must have the will to survive and they will attain happiness. This belief is passed on to Shinji, first in his return to the physical world in Episode 20, and later in his choice to be "alive" (existing as human in physical form with an AT Field) rather than accept Instrumentality. Yui's beliefs may also be exemplified in her romantic relationship with Gendo, however Gendo's desire to be associated with Seele raised questions concerning whether he was truly in love with Yui or whether he was using her to get closer to SEELE. The fact that his every action for the next decade was made in pursuit of reuniting with Yui suggests he was not just using her; conversely, since she kept some of her agenda secret from Gendo, she may in a sense have been using him.
In episode 21, Yui describes Gendo as "...quite a sweet person. It's just that no one knows it."
Yui has her own agenda, different from Seele's, and ultimately different from Gendo's as well. Much , but not all of her agenda is presented in the series.
After Yui (presumably) graduated, she married Gendo, and the latter took on her surname He became involved with Seele, joining the Katsuragi Expedition to serve as an auditor, although it is not revealed what Yui was doing during this time. After Gendo returned, they had a child together, Shinji who, if a girl, would have been named Rei.
She died in 2004 as a result of her Contact Experiment with Eva-01, but her spirit lives on within the Eva as eternal proof of humanity's existence.
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Neon Genesis Evangelion tackles heavy topics such as human identity, ambition, desire, regret and self-worth. Characters in the series are deeply flawed and spend a lot of time reflecting on personal issues. As such, the show is a goldmine of quotes on the subject of what it means to be human.