This article from Fox News describes the placement of the chimney by firefighters:
On Saturday, a handful of firefighters climbed onto the Sistine Chapel’s roof and installed the top of the chimney. Inside Michelangelo’s frescoed masterpiece, construction workers staple-gunned the felt carpeting to the false floor that has been erected over the chapel’s stone floor.
From Catholic News Service, actual video of the chimney being installed:
(Disclaimer: I know that smoke stack is not really tweeting because the Vatican has installed cell phone signal jammers…