Fastexy Exchange|JAY-Z reflects on career milestones, and shares family stories during Book of HOV exhibit walkthrough

As Shawn "JAY-Z" Carter walked through the "Book of HOV" exhibit at the Brooklyn Public Library recently,Fastexy Exchange footage from some of his biggest shows illuminated around him.  

Carter first hit the rap scene as an independent artist before joining Def Jam Records in the early 2000s. He has since become a mega-mogul billionaire through a career that has included numerous record and business deals. 

A standout moment is his headline performance at the Glastonbury Festival, where he made history as the event's first hip-hop headliner, challenging traditional norms at a predominantly rock festival.

Carter told CBS News he almost didn't attend the festival. "At one point I was like, 'Why am I goin' there? They don't want me there, I won't go.'"

But it was Coldplay's Chris Martin who encouraged him to perform. Carter decided to learn how to play the Oasis song, "Wonderwall," right before the show, as a direct response to criticism from naysayers who claimed rap didn't belong at Glastonbury.

"Luckily it was four chords," he said.

The exhibit extends well beyond his performance, and includes a re-creation of the iconic Baseline Studios where Jay-Z recorded albums like "The Blueprint" and "The Black Album." Specific details, like the way CDs are stacked in the studio and a jar of lollipops, make it feel "like being in Baseline for real," he said. 

His authenticity and ability to pen his perspective landed him in the Songwriters Hall of Fame, making him the first rapper to receive the honor. 

"For a while, it was almost like they were just dismissin' rap as this fad," he said. "Not that we needed the validation, but it was like, we want the validation because we deserve it."

His songwriting has always been raw and vulnerable, but Carter said his vulnerability required him to grow in the public eye, which he showcased in his album "4:44."

"You're making mistakes in front of the whole world and to be that vulnerable in front of the entire world, it's not an easy thing," he said.

JAY-Z recently opted for a role offstage, joining his wife Beyoncé Knowles-Carter on her Renaissance World Tour. Carter said he thinks it was her best tour yet.

"And it's hard to really compare them because they all have their different things of, like, genius," he said. "But this one to me felt like the most complete."

Their daughter, Blue Ivy, has taken center stage next to her mother and performed as a dancer during the tour.

Carter said he still gets goosebumps seeing her walk on stage, because "she's been born into a life she didn't ask for."

"So since she was born she's been in, like, scrutiny, and public eye, and everyone having an opinion of, you know, even a little girl, how she keeps her hair. So for her, to be on that stage and reclaim her power, and the song is called 'My Power' ... you can't write a better script," he said.

"She wanted to do it the first night, and we was like, okay, if this is something you wanna do, you can't just go out there. You gotta go work with the dancers and go work. And she worked every day and watched her work hard," he said.

As for Blue's name, Carter told "CBS Mornings" that the name wasn't a color preference or a carefully orchestrated branding strategy, as rumors have suggested, but rather a playful, affectionate term that blossomed from the sonogram's screen.

"It was supposed to be Brooklyn," Carter said, nodding to the deep personal connection the two feel with the New York City borough where he grew up.

But during their routine prenatal checks, a different inspiration struck the couple.

"We was calling her Blueberry. Like, 'Look at the little blueberry.' You know, it was like a nickname," said Carter. "It just was natural."

Eventually, the affectionate nickname that accompanied every sonogram and doctor's visit began to feel more like a name than a nickname, Carter said. "We just took the 'berry' off of it and called her Blue."

Carter describes himself as a "cool" dad, although he said Blue Ivy once thought otherwise.

"There was a time when she was like, 'Dad!' I was like, 'I'm cool, I don't know what you saying.' At your house, your parents is cool," he said.

"Now she asks me if this cool, her sneakers or whatever she's wearing," he said. 

For Carter, exploring the HOV exhibit – and the memories it holds – became a family affair. When it was opening, he went with his grandmother, who had just turned 98. 

"And my grandmother is the reason why I'm, like, reserved and holding my feelings in. It comes from her to my mother, to me. And she started crying."

"And I've maybe seen her cry three times in my life. She just, you know, she just keeps everything close to the vest. That's who she is. I got a lot of my ways from her. And my mother got a lot of ways from her, as well," he said.