To reach Monomon the Teacher, you must first defeat Uumuu, a giant jellyfish boss. Uumuu is invulnerable to your attacks until Quirrel intervenes, creating openings. Focus on dodging Uumuu's electrical attacks and striking when vulnerable. After defeating Uumuu, you can access Monomon the Teacher's chamber and break her seal with the Dream Nail, removing the second Dreamer barrier.
