• Adductor muscles and gracilis are innervated by the obturator nerve
• External oblique is innervated by the iliohypogastric nerve
• obturator internus is innervated by the obturator internus nerve (L5–S2)
• Piriformis is innervated by the sciatic nerve (L5-S3)

"Innervates the muscles of the medial compartment of the thigh (obturator externus, adductor longus, adductor brevis, adductor magnus and gracilis)."

https://teachmeanatomy.info/lower-limb/nerves/obturator-nerve/

Obturator means to close; it closes (ie adducts) the legs.