Mr Bush had passed Mrs Obama a cough drop when they were seated next to each other at Senator John McCain's memorial service in September.
The loyal yellow Labrador returned to his faithful owner Tuesday, visiting the former president and his former handler's casket in the U.S. Capitol Rotunda after accompanying Bush's remains from Texas to Washington D.C.
James Baker, a longtime friend who served as Bush's secretary of state, eulogized the former Republican president as a peacemaker and "a truly lovely human being".
Another photo shared Monday showed Sully with Bush family members before the president's casket was taken to Ellington Air Force Base.
Bush died on Friday at the age of 94, he was president from 1989 to 1993, and will be buried in Houston, Texas, on Thursday.More news: Nation says goodbye to Bush: 'To us, he was close to perfect'
The founder of Poland's anti-communist Solidarity movement, Lech Walesa (lek vah-WEHN'-sah), attended U.S. President George H.W. Bush's state funeral in Washington wearing a T-shirt with the word "Konstytucja" (Constitution), a symbol of a political struggle in his homeland against the populist government. There will be no mail delivery Wednesday.
Fred Curry, one of the few African-Americans in line, is a registered Democrat from Hyattsville, Maryland, who voted for Bush in 1988, the election won by the one-term president.
One of the things that the 41st president liked most about his top diplomat was that "I'd always tell him what I thought, even when I knew he didn't want to hear it", Baker said, speaking at Bush's funeral at St. Martin's Episcopal Church in Houston Thursday.
Trump and first lady Melania Trump arrived by motorcade at Blair House on Tuesday.
Vice President Mike Pence said "President Bush was a great leader who made a great difference in the life of this nation".More news: Martin Solveig asks first female Ballon d'Or victor about "twerking" on stage
People filed into the Capitol throughout the chilly, overcast day to pay respects to the 41st president, a son and father of privilege now celebrated by everyday citizens for his common courtesies and depth of experience.
Bush was remembered just feet away from what he called "Democracy's front porch", the west-facing steps of the Capitol where he was sworn in as president.
"I think he was one of the kindest, most generous men", said Marge Frazier, who visited the downtown statue Sunday while showing friends from California around.
But as the rain began to fall everyone opened big black umbrellas - everyone except George W Bush and Jeb.
Flags on public buildings are flying at half-staff for 30 days.More news: Sprint on its 5G plans: You 'better believe' we're working with Samsung