In Jean Edward Smith's biography of George W. Bush, he wrote about what President Bush thought of President-elect Obama during the transition period.
On page 650, it states:
“As part of the presidential transition, Barack Obama asked Bush if it would be possible for him to meet all the ex-presidents. Bush was happy to oblige, and organized a White House luncheon in the Oval Office on January 7. Bush and Obama were joined by Jimmy Carter, Bill Clinton, and George H. W. Bush. The luncheon lasted over two hours, each former president ordered his lunch à la carte from the White House mess, and the tone was convivial and friendly. “All the gentlemen here understand both the pressures and possibilities of this office,” said Obama before the meeting. “For me to have the opportunity to get advice, good counsel and fellowship with these individuals is extraordinary, and I just want to thank the President for hosting us.”
Bush was equally effusive. “We want you to succeed,” he replied. “Whether we're Democrat or Republican we care deeply about this country. And to the extent we can we look forward to sharing out experiences with you. All of us who have served in this office understand that the office transcends the individual.”