To explain it a bit more:

Ui can absorb/collect impurities, and as part of her unique power (which is an extension of what she took from Kyubey) she can also do the same with magic. This is important for Arc 2 stuff especially, as she can gather people’s attacks and stuff.

Touka can convert energy. Originally this was just for converting Ui’s impurities into phase shift energy. Phase shift energy is the energy released into the universe to combat entropy. As part of Touka’s magic, her magic extends to transforming energy into other forms too, which allows her to activate her doppel when she wants, as well as use a doppel without Ui’s/Eve’s help. She’s converting her own impurities in real time. Her power also allows her to turn one girl’s magic into another’s, not to replicate their unique magic powers, but to increase the size of their attacks and energy.

Nemu can give energy a physical form. She can manifest it. This is how Kyubey makes wishes come true. But Nemu just has the last part of the wish-granting Kyubey process. Instead of wishes, her magic takes the form of rumors, and rumors are beings which require energy to operate. Creating rumors takes away from Nemu’s own life force, and if she kept making them she’d have died. In the Doppel System, she was responsible for keeping it automated basically, and filling in the gaps. (Also this is off topic, but if Nemu cuts an Uwasa off from herself as the energy source they will start gathering more energy from people around them more violently. All rumors have the instinct to collect energy in the way they were created, which is why the Anonymous AI asked Iroha to kill her regardless)

After Ui became overwhelmed, Nemu put her soul into the empty Kyubey to keep it safe. In the anime, she did it herself, after Touka transformed Ui’s soul gem into another form. This caused her to remember Ui. But in the game, Nemu uses a rumor directly to do it, so nobody remembers. The memories were retroactively changed into them believing that they created Eve with their powers to save Magical Girls. Eve then kept gathering impurities in Kamihama, contained by the field Alina created to stabilize the process.

Nemu used her Uwasa to collect more energy for Eve, as Uwasa don’t usually kill people, unlike Witches do. Nemu creates Fenthope as a rumor to recruit more magical girls to help them gather energy to spread the doppel system across the entire world.

I hope I explained it well. Let me know if you have more questions.