On day 6, Devi Kātyāyanī one of the avatars of Mā Śakti is worshipped. As per Śrimad Bhāgvata Pūrāṇa, young eligible ladies in Gokula observed fast during the month of Mārghaśīrṣa and offered prayers to Goddess Kātyāyanī to get Bhagwān Kṛṣṇa as their consort.

Mā Kātyāyanī is one of the highly revered forms of Mā Durgā signifies cheerfulness and joy. Believed to be the destroyer of all evil, she is seen as a warrior goddess who was able to bring peace to the world.
As per Pūrāṇa, long ago there was a Riṣi Kātyāyana. He was a stout devotee of Goddess Śakti. Riṣi Kātyāyana always wished for Devi Śakti to be born as his daughter.
During this period, an evil demon named Mahisāṣura was creating great trouble for the Gods. With time he gained more power and the Gods were increasingly worried about him. They prayed to Devi Śakti to save them from the wrath of Mahisāṣura.
Devi Śakti decided to take birth to end the reign of Mahisāṣura. She granted the wish of Riṣi Kātyāyana and was born on Earth as his daughter. She grew up to be a strong, beautiful fighter and came to be known as Kātyāyanī.
One day, Chandhā and Mundhā, two messengers of Mahisāṣura, saw Kātyāyanī and were mesmerized by her beauty. They went back to their master and told him all about Kātyāyanī. Mahisāṣura became happy and asked Dundubhi, another messenger, to go and talk to Kātyāyanī.
Dundubhi met Kātyāyanī and told her all about Mahisāṣura and his greatness. He told Kātyāyanī that she should marry Mahisāṣura as they would make a great couple. Kātyāyanī smiled and told Dundubhi that according to her family tradition, Mahisāṣura must first defeat her in battle. Only then can they both get married. Dundubhi went back to Mahisāṣura and told him about this challenge. Mahisāṣura agreed and soon the preparations began.
A fierce battle took place between Kātyāyanī and Mahisāṣura. She managed to defeat his demon soldiers and even killed most of them. When Kātyāyanī and Mahisāṣura came face to face, he transformed himself into a buffalo. This proved to be a real challenge for Kātyāyanī as she struggled to fight him.
However, Kātyāyanī tricked Mahisāṣura and climbed onto his back. He was surprised by this move and tried very hard to shake her off, but failed. Mahisāṣura then placed her foot on the back of his neck, pierced him with her Trishul (trident), and cut his head off.
This is how Mā Kātyāyanī killed the evil and powerful demon Mahisāṣura. In doing so, she saved the Gods from his menace and brought peace back into this world.