Disclaimer: I do not own Inuyasha or any other characters (except Katsurou).
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Sesshoumaru realized that convincing the child's mother, might be more difficult than he imagined.
"This Sesshoumaru did not realize the extent of the problem. Obviously, Katsurou has abilities that need training in order to be in control. He cannot get the help he needs from you or anyone else. If he becomes my son, he will be protected in a world that does not understand him yet. He will not be welcome among humans or demons. As my heir, I can fully provide for him."
"He is not, nor ever will be your son! He is Inuyasha's son! You cannot replace him!" Kagome raged.
Sesshoumaru was at a loss for words. How could he convince her that it was in their best interest? She was obviously still grieving her husband. She wanted to keep Inuyasha's memory as a father alive and well in Katsurou. This, he could understand.
He was fighting an uphill battle, and losing quickly.