How do you get the nemesis in Cave Story?

Go to the spot where Kazuma's Dragon is. There is a small overhang there. Go into it and find a door. Talk to the very tiny woman. She is looking for her husband. Go to the Mimiga Graveyard and talk to her husband, he is very small, look for tiny moving pixels. Return to his home. He will then allow you to trade the nemesis for your blade.