WARNING: The following contains spoilers for Episode 3 of Yashahime: Princess Half-Demon, "The Dream Butterfly," now streaming on Crunchyroll, Hulu and Funimation.
This week's action-packed episode of Yashahime: Princess Half-Demon finally reveals the familial identities of the three heroes in a more or less concrete way (though things are still somewhat up in the air for Moroha). Thanks to the return of Root Head from "Inuyasha: Since Then," there's also an explanation for the time travel that has appeared thus far throughout the anime.
Much of "The Dream Butterfly" is made up of battles, first between the trio and the centipede demon and then between Towa and Setsuna. However, the most important moment of the show is the revelation of the twins' parents. They are the daughters of Inuyasha's demon half-brother, Sesshomaru. Kaede explains this, speaking to the demon hunters while they report on the vanishing of Moroha and Setsuna. She also reveals her opinion on why Sesshomaru left them to their own devices in the woods for all those years -- the rite of courage and cowardice.
According to Kaede, a lion will intentionally leave its cubs in the ditch to see if they crawl back up. Basically, a cub is only worth raising and caring for if it proves it's capable of surviving. This is how Kaede looks at Sesshomaru and his treatment of Towa and Setsuna -- he tested them by abandoning them, to see if they were worthy of his fatherhood.
However, this interpretation doesn't exactly jive with Towa and Setsuna's idyllic childhood, and as Kohaku says, it is always difficult to understand Sesshomaru's true intentions. There's further reason to doubt Kaede's interpretation of things as she says when asked about Towa's absence, that it's entirely possible Setsuna killed her sister out of inherent demonic hatred. Quite to the contrary of Kaede's reading, these two had an incredibly strong love for each other and were driven apart by chance. Given Kaede's mistake here, it would make sense she might also be wrong about Sesshomaru, who might have more complicated motivations than just testing his daughters.
In the present, however, the sisters are far from on good terms. Throughout the episode, they fight, though admittedly Towa is possessed by a demon for much of this time. Still, Setsuna claims to not remember Towa, and for the time being rejects the possibility of a relationship. She seems much more connected to her demon side than the more human Towa, something that will surely lead to conflict in the coming episodes.
Moroha, meanwhile, is revealed to most likely be the daughter of Inuyasha and Kagome. Great-Grandpa immediately spots the resemblance, and later on Sota also notes how much Moroha resembles his sister Kagome. Moroha is still the character viewers know the least about. She is clearly fun-loving, fearless and confident to a fault, but her past and how she came to be a bounty hunter is still a mystery. Still, expect to learn more about her soon, particularly as next week's episode will focus on her attempts to return to the past.
This episode also explains the time travel that takes place in Yashahime. That Root-Head, the demon from Episode 1, has embedded a piece of himself in the Tree of Ages. As it turns out, the Bone Eater's well from Inuyasha was made from wood from the Tree of Ages, along with its magic ability to open portals through time. Root Head, having embedded himself in the tree, is capable of manipulating it. What this means for the future remains to be seen.
Yashahime: Princess Half-Demon has been accused, quite fairly, of being a difficult show to jump into. However, things are starting to get clearer as the show reveals more about the characters' origins and the world they inhabit. With this background info accounted for, all that remains is for the heroes to find a way to work together and get back to the past.