Is the hat tohru has kyou's?

Yes it was first Kyo's, shishou (his "dad") bought if for him when he was a little kid. The wind blew his hat off when he was taking a walk, then his enemy Yuki tried to give it back to him he wouldn't take it because Yuki touched it. Then Yuki sees Kyoko (Tohru's mom) Freaking out and she says to some police officers that her baby (tohru) Is missing so Yuki remembers seeing her in an alleyway as he was walking by. He runs back to the alley and sees Tohru crying, she gets up and follows him and she gets to her mom. Kyo's hat that Yuki was wearing falls off as he turns into a rat and Tohru kept it all these years and doesn't know that yuki was the boy that saved her!