Fair warning, I haven't edited this and I wrote it on my phone. There may be typos. This was inspired by, dare I say it? Twilight and the 'imprinting' the wolves have. I thought it was interesting and added to why the Inu no Taisho would leave his mate for Izayoi and why Sesshomaru would allow Rin to follow him, as well as explain why Inuyasha is stronger when he's protecting Kagome.
"Milord, we need to discuss what will be done with Rin," Myoga said once they were alone. "Arrangements need to be made."
"She will stay with me," the taiyoulai replied as though the answer was obvious.
"Milord, she can't remain with you forever."
"She is a child, where would she go?"
"She is a child now, but she will grow up and soon. In a decade she will be ready to mate and bear young. What will you do then?"
"She will stay, Rin has made it clear she wants to be where I am."
"So it happened to you too," Myoga sighed. Sesshomaru frowned, waiting for the flea to elaborate, when he remained silent Sesshomaru sighed and asked.
"What has happened?"
"You have found your mate," Myoga explained, a surprised tone in his voice.
"Nonsense, there is no mate."
"Your soul mate," Myoga corrected. "Your father believed that demons were meant to protect humans. What he didn't know is that a demon is meant to protect one human, a human whose soul is mated with their own. It is why taiyoukai have such long lives, it can take centuries to find their mate. When they do however any offspring are more powerful, regardless of their hanyou status," Myoga explained. " Your father's was the Lady Izayoi, Inuyasha had the misfortune of having his mate's soul reside in two bodies; Kikyo and Kagome. Your mate is Rin."
Sesshomaru pondered what the flea demon had said. It explained more than it didn't, he still didn't see the purpose of the private discussion however.
"This news changes nothing, if anything it gives more reason for her to stay. Once she is ready we shall be mated and she will bear strong pups."
"She will grow up, Milord it's true, but she will also someday grow old. You will not have her forever. She has already been nearly lost twice. Could you bear to lose her after she had mothered your young? To see her grow old in a matter of decades while you look the same?"
"So we've come to the reason for this audience."
"Milord you must make a choice, you can leave her with her own kind and go your own way or you can be with her while she lives her own short life."
Sesshomaru left her in the care of Kaede while he traveled, though his frequently visited. He never said where he went or what he was doing, after a while people stopped asking.
For years the Western Lord searched for a way to keep Rin alive, for her to stay young with him. Myoga had been right, the dog demon did not want to see her grow old beside him while he remained unchanging.
Finally the time came, Rin had reached her eighteenth summer and it was now or never.
The old flea demon waited under the tree Inuyasha had been pinned to not so long ago with Rin. It still surprised him how quickly she had grown. It was as though she had been a child with a missing tooth and flowers one moment and in the blink of an eye she was a grown woman. He would need to remind himself not to blink anymore.
"Milord, I take it you've made your decision?" Rin looked up at him, her dark eyes so opposite his golden ones.
"It is not just my decision to make," he replied softly, eyes still locked on Rin. "Do you wish to remain here with your own kind, or be with me?"
"Lord Sesshomaru you know the answer, where ever you go I wish to go too. I don't care if I grow old and die, as long as my years are spent by your side." His eyes widened, how could she have found out what he was trying to do? And to choose to be with him even if it meant ageing and dying?
"Well Milord?" The corners of his mouth turned up slightly, a ghost of a smile on his face. If she could place such blind trust in him, he could trust her. Sesshomaru hadn't found what he had been searching for, but he found something that would suffice.
"I choose a mortal life."
So yeah, Sesshomaru couldn't find a way to keep Rin young, so he gave up his 'immortality'. He will age at the same rate as her now, though he will still have his demonic appearance and powers. Their children will be hanyou like Inuyasha.