Emi: You... Will you wait for me? When I'm old enough I'll marry you. So... will you?
Yagi: Heh, if that's what you want ok! I'll wait for you. Promise!
Emi: Heheh promise!

This is the moment Emi first was saved by Yagi. Love at first sight for her. And the promise was made, one she hold on to and drives her to keep going. To meet him and fulfill their promise.