updated 06/16/2013 AT 3:00 PM ET
•originally published 06/15/2013 AT 8:15 PM ET
The controversial rapper, 36, was “supposed to attend his listening session last night,” a source tells PEOPLE of his new album event, scheduled for 10:00 p.m. on Friday. “Kanye canceled about 15 minutes prior [to the playback] to be by Kim’s side.”
Before officially calling off the session, West was spotted much earlier on that day with Kardashian, 32, who “looked very exhausted and tried to be incognito,” an onlooker, who Tweeted about the Cedars-Sinai Medical Center sighting, tells PEOPLE. “Kanye seemed very supportive. It looked like support in pain and labor.”
The Keeping Up with the Kardashians star’s E! network confirms West was with her for the birth, adding that the couple hasn’t revealed their daughter’s name yet, but that Kim told friends it “will start with a ‘K,’ ” and the middle name could be Donda, after West’s late mother.
West – who played his new album, Yeezus, for fans at Art Basel in Switzerland earlier this week – likely wasn’t the only guest in tow to deliver well wishes to the new mommy. An eyewitness spotted longtime Kardashian pal Brittny Gastineau and her mom Lisa on Saturday at the Los Angeles hospital.
Reporting by JESSICA HERNDON, DANA KENNEDY, RAHA LEWIS and GABRIELLE OLYA
[RELATED_TEXT “More On Kim Kardashian: