SataYuri

SataYuri is the het ship between Satan and Yuri Egin from the Blue Exorcist fandom.

Manga
Back when people first knew Satan as Rinka, Yuri came across Rinka's blue colored flames as a child and played with the demon, who didn't know that they were playing at the time, by lighting the items Yuri has set up for him on fire.

Anime
The second part of the first season anime doesn't fully follow through with original manga, along with the anime's version of Satan and Yuri's story airing long before the manga revels it.

When Yuri was sent to investigate spontaneous human combustion with a team of fellow exorcists, she was the only one to survive Satan's attempts to possess them as well as her hearing Satan's voice before passing out. Curious about the voice and blue colored flames Yuri stays behind in the forest to seek them out, where she comes across Satan who is possessing the body of a white furred wolf, that was about to be communed and destroyed his flames. As he threaten to hurt Yuri she allowed him to bite her leg without any resistance from her, and after hearing Satan say that his flames prevent him from staying in Assiah until the bodies he possessing give out, Yuri takes pity on the demon and willy allows him to use her body.

During the year Satan possessed Yuri's body since that winter, she made the forest her new home so she can show Satan the beauties of her world through her eyes. As well as teaching him several things about Assiah, in which allows Satan and Yuri to become quite close with one another. One day, Yuri finds a baby bird that had died from falling out of its nest Satan asks her what she means by life and death. Yuri's words on the beauties of life has Satan wanting to have a life, but Yuri explains that it isn't someone that someone would take or buy for themselves, so she recommends that they could at least nurture one Yuri gets impregnated by Satan. At one point, before or after Yuri teaching Satan about the beauties of life, they talk about a world for both humans and demons living as one, and because no such world exists Yuri recommends that they create that world as she asks Satan to close his eyes so he could picture that world as they dream about it together. As Yuri places her hand in the water of her reflection to make it feel like she and Satan are holding each other's hands.

Yuri kept Satan and her pregnancy of their child a secret from Shiro Fujimoto when he was sent to take her back to the Vatican by (the anime version of) her father, Ernst Frederik Egin. Until Shiro shooting at her demon friends provoked her into revealing Satan's flames before passing out, and when she regained conscious she confesses to Shiro about letting Satan in and having no shame about being pregnant with the demon's child. Ernst on the other hand was disgusted that his only daughter would allow Satan into her heart and wanting to have his child, he disowns Yuri and calls her a witch before putting her into court where she was sentenced to death as a witch. In order to save Yuri from her fate, he searched Assiah for vessel that would allow him to do until he possessed Ernst and told Yuri to run after using her father's body to cut her free.

Even though Yuri loved their twin sons, Rin and Yukio, Satan wasn't very kind towards the Okumura twins. When Satan possessed Yukio, after he tells Yukio of truth of what really happened to Yuri, the Demon God decides to the just Gehenna Gate to make his and Yuri's dream world come true and guards the gate to make sure that Rin doesn't destroy it again. As Assiah and Gehenna were slowly begging to merge with one another, Satan looks back to the day he and Yuri had first talked about their none Assiah and Gehenna world. After Rin and Yukio destroyed the Gate to save Assiah, Satan wonders why their sons would want to stop him from creating that world, to which has Yuri's spirit telling Satan with a smile that he was in a rush to make it a reality and that they should leave the creation of their human-demon world to their sons.

Children
Rin Okumura

Rin is Yuri and Satan's elder twin son who has inherited Satan's flames, which is why Yuri decided to name him Rin as she first knew Satan as "Rinka" and got along with Rinka well before he became to change as Satan. For a long time, Rin didn't know anything about his birth parents or that he is half demon, until he became to see demons and guardian, Shiro, knew that it was time to tell Rin the truth of who his biological is. Satan possessed Shiro in an attempt to bring Rin to Gehenna, Shiro kills himself to prevent the demon from taking Rin away, while Rin unseals the demon slaying sword that had sealed away his demon half from his human body, which returned it to Rin, before destroying the Gehenna Gate. Favoring his caring adoptive father over his cruel biological father, Rin swore that he'll become an exorcist, like Shiro was, in order to defeat Satan.

Yukio Okumura

Yukio is Yuri and Satan's younger twin son who didn't inherited Satan's flame, which made him fully human. Even though Yukio has been apart of the exorcisting world since he was little after becoming an exorcist as a child, he knew nothing about his mother or that Yuri was an exorcist herself, he didn't however learn that Satan is his father and not inheriting Satan's flames is what allowed him to become an exorcist and be expected by them; while his elder twin brother being born as the demon sibling of the pair places a target on Rin. After facing a man named Saburota Todo, Yukio learns that he too has inherited a much smaller part of Satan's demonic power through his eyes.

Fanon
Long before the story of how Satan and Yuri met and had their twin sons was revealed in the original manga, the anime releases its own version of their story which inspires fans to support their relationship, as well as some comparing them with Sparta and Eva from Devil May Cry since they too have half demon-human twin sons of their own. The ship's support grew after the manga finally reveals its version of their story, along with a few fans, as well as Yuri herself, being heart broken when Satan's rage and obsession with Yuri communed him, along with the True Cross's actions making Satan believe that Yuri betrayed him for Shiro.

In fanon, that is first based on the anime version, fans have the two rising Rin and Yukio together, one way for another, and had featured Satan with his own human body so he and Yuri can stand side by side each other.

On AO3, the ship only has 13, while fanfiction.net has 24 with only 5 placed under pairing.

Fandom
FAN FICTION
 * Satan/Yuri tag on FanFiction.net
 * Satan/Yuri tag on FanFiction.net