What was the story of Demeter Persephone and Hades?

Persephone is Zeus and Demeters daughter. One day while in the meadow with her mother, picking flowers. Hades was nearby bringing dead souls to the underworld. Eros, the god of love, shot an arrow at Hades and he instantly fell in love with Persephone. The first person he saw. He then raced up in his chariot of black horses and captured her to bring down to th underworld. Demeter, the goddess of harvest, did not let anything grow until she found her daughter. Then Demeter had a idea, she went down to the underworld and found Hades with persephone, and demanded her return. Hades, with no other option had to let her leave. but before she did he had her eat a few seeds (if you eat in the underworld u will have to come back) and one day Persephone wanted to go back and now she lives with Hades as king and queen of the underworld. persephone only lives in the underworld during winter and autumn, she is with her mother in the spring and summer. that's why there are seasons