NASA funded lander built and operated by Intuitive Machines, launched on a SpaceX Falcon 9, delivering NASA scientific payloads.

There are several! Instead of funding one large, expensive NASA mission, they took the unusual approach of funding many small and new companies a relatively small amount of money to develop their technology and attempt a landing with some NASA payloads under the Commercial Lunar Payload Services (CLPS) program.

The idea is that they know some are going to fail, especially their first attempt. The NASA administrator in charge of the program has described the strategy as “shots on goal.” These are basically startups with untested technology using “cheap” stuff instead of traditional aerospace materials because that’s all they could afford. But the payoff could be huge if they do actually succeed and they’re getting closer and closer to that! Here’s a rough schedule:

https://www.planetary.org/space-missions/clps

So yes, these are all private companies that received NASA funding to help develop their landers, but the funding was small for a mission of this scale and most the companies bid at a loss in order to win their missions.

My understanding is it's a NASA mission but they hitched a ride on a private rocket