Morningside Heights is part of greater harlem. https://en.wikipedia.org/wiki/Morningside_Heights,_Manhattan
The distinction between Morningside Heights and Harlem is a linguistic fiction and largely in name only. It’s mostly used as a tool to separate Barnard/Columbia from the low income community that surrounds it. It has historically been part of both the UWS and Harlem’s congressional districts. Saying Columbia is in Morningside Heights and saying it is in Harlem are pretty much the same thing. The only difference is that most people have heard of Harlem so it is more effective in describing it’s location to people who aren’t as familiar with NYC’s very specific neighborhoods.
Columbia is also deeply entrenched in the Harlem community as much of its community outreach centers on harlem and the community groups with which the University must most closely work during expansion are Harlem groups. And students are MORE than happy to socialize in harlem given the awesome restaurants, bars, parks and more.
But yes, Columbia is technically in Morningside heights. Though anyone that’s ever been to Columbia knows you could describe it as both being on the (very) Upper West Side and in Harlem. It’s overly nit-picky to say otherwise.