YashaHime: Princess Half-Demon is a 2020 anime sequel to Inuyasha which originally ran in Japan from 2000-2004 with the English Dub running on Cartoon Network's Adult Swim from 2002-2006.
The first episode of YashaHime (Titled: Inuyasha: Since Then) picks up six months after the conclusion of the original series and introduces the audience to the everyday lives of the original main characters. Miroku and Sango are happily married with three children. Inuyasha is working with Miroku as a pseudo-demon-slayer, and Kagome is studying to take her place as the village priestess.
Ok, so far so good.
The episode itself uses an intro technique known to writers and literary critics as a mini three-act structure. This type of intro is, by definition, a complete story on its own, containing a beginning, middle, and end, with the central conflict serving as an exemplar of the type of conflict the audience can expect going forward. The problem with it in this instance though, is that the conflict used in this episode (the battle with Root Head), actually doesn't have much to do with the plot going forward. For a mini three-act structure to work well, it needs to do two things:
1. Offer a representative of what kind of conflicts the viewer can expect and how those conflicts will be handled.
2. It needs to have something to do with the conflict in the main story, it could be a pivotal scene that the main characters couldn't possibly be aware of, or introduce a mystery that hooks the viewer.
The first episode of YashaHime half succeeds at the first point, what we're shown is a group of friends/family slaying demons together, and that is what the majority of the conflicts are. Where it fails is that it sets up the expectation that the majority of the slaying will still be carried out by Inuyasha, Miroku, and Sango, just like in Inuyasha itself, but that's no longer the case after this episode.
As far as the second point is concerned though, not only is it an utter failure, with Roothead (the demon they slay in episode one) having nothing to do with the central conflict of the main story, in fact, he is killed yet again at the end of YashaHime episode 4.
Furthermore, there's a much better prologue in the first few minutes of episode 2. In it, we see two of our three real main characters living alone in the forest despite the fact that they're extremely young. Where are their parents? One of them ends up being thrown 450 years into the future, how did that happen? Then we have the introduction of the third of our main trio and find out that her parents are missing as well, where are they?
These are questions that have to do not only with season 1's plot but will apparently be delved into more deeply and possibly answered in the upcoming season 2, in addition, they were guaranteed to hook a lot of the old fans as well as flesh out the new conflict with our new main antagonists.
See how much better that serves the purposes of a prologue than episode 1?