Crossovers that MUST be

[Jorlem]

And later on Hayate closed time loops it to her younger self?

Not a closed time loop - more an S-curve in its temporal track. The Book of Darkness ends up being a decade older than it "should" be.

The way I was thinking, the Book itself isn't a closed time loop, but Hayate's association with it is. The serial killer used the Book as a focus and summoned Hayate, because she is associated with the Book. Hayate is associated with the Book because she left it with her younger self, thus ensuring the chain of events that would allow her to be summoned using the Book as a focus.
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Stand between the Silver Crystal and the Golden Sea.
"Youngsters these days just have no appreciation for the magnificence of the legendary cucumber."  --Krityan Elder, Tales of Vesperia.