As you can see in the image above, Summoned Symbiote cannot be upgraded with normal Crystals or Catalyst. You need Klyntar Katalyst to upgrade him. So where to get them?
The only place to get Klyntar Katalyst is “Cosmic Elipse” Side Quest.
But, how do you Enter Cosmic Elipse Side Quest?
Kabam gives you a special solo objective every day. You must complete that to receive the Entry Token. That Entry Token is valid for 1 entry. Just keep repeating that every day: Compete Solo Objective and Enter Cosmic Elipse Side Quest to get the Upgrade Material.
Which Quest Should I Enter in Cosmic Elipse as there are 6 of them?
There are 6 quests in the Cosmic Elipse Side Quest. You can enter any of them and they will offer 10 Kalyntar Catalyst. However, there are some additional rewards for each quest award depending on which one you Enter. Before you enter the Quest, there is an option “View Quest Rewards” on the bottom left. Click on that option to check what reward you will get before you enter.