Inuyasha is a bit(more like a lot) out of character in these last couple of pages but he'll be back to the swearing, loudmouth dog hanyou we all know and love pretty soon. ^_^
Basically he babysits Muteki in his spare time 'cause Sesshomaru has to work like 24/7 with his new agency.
Muteki's the reason why Inuyasha is pacing forward in life even though his newborn daughter drowned in the sea and his beloved wife(Kagome) disappeared along with her.
He's almost like a son to him.