I have just closed another thread you opened, as in that question you stated you were using Pop OS! 22.04 and had added the MATE desktop on top of it.
Ubuntu MATE 22.04 isn’t on-topic here either; as being a flavor of Ubuntu, it comes with 3 years of support for LTS releases; please refer to the Start Here page.
You should have no problems adding the Ubuntu MATE desktop onto a Ubuntu Desktop system, or vice-verse either, as I’m a lover of multi-desktops myself, and have thus done it. But from your prior question where it wasn’t Ubuntu you stated, but Pop OS! that you also referred to as Ubuntu in the same thread (it’s not; Pop OS is Ubuntu based only), I’m closing this thread too.
If I’ve misunderstood, or you wish to clarify, you can PM me (or another moderator) with clarification/explanation, edit your details and we may re-open this thread.