Jim McGrath, Bush's spokesman, posted a picture next to Sully next to Bush's coffin on Sunday entitled "Mission Complete."
Sully was named after former flight pilot Chesley B. "Sully" Sullenberger III, who became famous when in 2009 he landed a damaged passenger plane on the Hudson River and rescued all 155 passengers and crew.
Sully is a well trained companion dog who will now help other veterans and will go to Walter Reed National Military Medical Center, former President George W. Bush wrote in an Instagram post.
"As much as our family will miss this dog, it is comforting to know that he will give his new home, Walter Reed, the same enjoyment that will be brought to 41
Sully can execute a two-page list of commands, including answering the phone and retrieving items.
Sully went to work with Bush this summer after former First Lady Barbara Bush passed away earlier this year.