Altering the history of the Over-Time Stream.
Especially altering history that had already solidified long ago was like forcibly chiseling open a concrete road that had already set.
No one wanted to accomplish this more than the Anti-Monitor. As long as he could truly alter the history of the Over-Time Stream, he could free himself from the failure of the Infinite Earths Crisis and let the Multiverse witness the magnificent power of the Anti-Monitor at his peak once more.
But the Anti-Monitor had never truly succeeded.
The Multiverse was simply too powerful. The DC Multiverse was no ordinary multiverse. At the moment of its creation, Mother of the Multiverse Perpetua had created DC as a weapon—a weapon to fight the other members of the Source.
The power of the Multiverse was not something a single Sixth-Dimensional God could oppose. Even the Anti-Monitor at his peak, after devouring countless positive-matter universes, could not do it.
He was indeed powerful enough, but not powerful enough to do whatever he pleased.
Therefore, even the Anti-Monitor genuinely longed for this to happen:
For someone to get there before him and help shake the entire Over-Time Stream.
To create some small changes in the Over-Time Stream first.
For example, taking Earth-15 away first.
First Lantern's success would also benefit the Anti-Monitor.
Once his plan was complete, Earth-15 would be taken away from existence. The history of the Emotional Spectrum and the Green Lantern Corps would continue to remain on the Over-Time Stream, but that road would probably already be riddled with cracks by then.
Simply put, First Lantern's actions would cause history to suffer even more damage, making it easier for the Anti-Monitor to shake his own history.
The Anti-Monitor and Batman Dragon had fought over Crisis Energy simply to descend from the Over-Time Stream