Okay, I'm going to walk you through what happened while playing, and see if someone can help me.
I am defender
One attacker declared, one defender declared
Edge Battle occur, I play a Fate card that deals one damage to a participating unit
I damage and destroy his one attacking unit
So now I have my defender unit, with no attackers. What can I do with it? There's nothing else participating in the conflict, can I attack a non-participating unit? Can I attack one of his objectives? Or do I just not get to do anything since there's no participating unit?