“Lili becomes the fourth generation of amazing, strong women in the family—behind Meghan, Diana and her Majesty the Queen,” says the couple’s friend, Dean Stott

Meghan Markle and Prince Harry are now settling into life as a family of four after a whirlwind year that saw them embark on a new life in California.

The proud parents, who have said that their daughter Lili "completes" their family alongside 2-year-old Archie Harrison, "are so happy," a close friend tells PEOPLE in this week's issue. "This baby solidifies that they are creating roots here in the States."

Although they stepped back from their roles as working royals, they have always said they never left the royal family itself. And when they announced the June 4 birth of their daughter, the baby girl's name—Lilibet "Lili" Diana Mountbatten-Windsor—spoke volumes.

The name Lilibet — Queen Elizabeth's childhood nickname — signifies the closeness that endures between Harry and the Queen, says royal historian Robert Lacey.

Prince Harry and Meghan Markle
Credit: Samir Hussein/WireImage

After the 1997 death of Princess Diana, "She was a sort of stepmother for both Harry and William," says Lacey. "We have all focused on her coaching of William as the future King, but we are realizing now the importance of the emotional bonds that she's been able to establish with Harry, and their ability to talk to each other directly. That affection remains."

Prince Harry, Duke of Sussex and Queen Elizabeth
Queen Elizabeth and Prince Harry
| Credit: Yui Mok/WPA Pool/Getty
Prince Harry and Meghan Markle Royals Cover

After returning home from the hospital, they introduced Harry's grandmother to her namesake via a video call, according to a source: "They were very excited and couldn't wait to share that their daughter arrived."

As they embrace their new status as a family of four, they plan to concentrate on life at home for awhile. Big brother Archie "is very happy to have a little sister," says a spokesperson for the couple. And Meghan and Harry are adjusting to the newborn routine again, only this time, they are doing it on their terms.

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Archie Mountbatten-Windsor 1st birthday
Meghan Markle and Archie
| Credit: The Duke of Sussex/@SaveChildrenUK

In keeping with the 20-week paid parental leave policy of their Archewell Foundation, both Meghan and Harry plan to take several months off to spend time with their daughter.

Listen below to our daily podcast PEOPLE Every Day for more on Prince Harry and Meghan Markle's daughter, Lilibet Diana.

"Lili becomes the fourth generation of amazing, strong women in the family—behind Meghan, Diana and her Majesty the Queen," says the couple's friend, Dean Stott. "Harry and Meghan now have their complete family. It's their time to be in the moment."