(CBS News) It's the moment \"Late Show\" fans spent the summer waiting for: Stephen Colbert debuts as host Tuesday night after moving to CBS, reports CBS News correspondent Vladimir Duthiers.

For nearly a decade, Colbert entertained his cable audience as host of Comedy Central's the \"Colbert Report.\"
Now, he's entering a network, late-night landscape that he helped transform, one much more crowded than it once was.

\"There are many more voices vying for attention. Late-night is not necessarily the big tent that it used to be under Johnny Carson, and really more an environment in which one wants to establish a distinctive voice,\" New York Times chief TV critic James Poniewozik said.

Colbert had that distinctive voice. His character on the \"Colbert Report,\" also named Stephen Colbert, was, in his words, a \"poorly informed, high-status idiot.\"

It was a parody of a right-wing political pundit, funny to many, but not what he can or will bring to CBS. He shared some of his vision of the new show with \"Sunday Morning's\" Mo Rocca.

\"The goal is to have fun with my friends, and sometimes that means talking about what you care about. We're going to be talking about what's going on in the world,\" Colbert said.

Broadening the audience will be key. Colbert takes over for David Letterman, whose late show has ranked second to the \"Tonight Show\" since 2009.
\"I think he's expected to and wants to bring over much of his audience, which is younger than the other Late Night audiences in the same time slot and I think that is a big reason why CBS wants him,\" Poniewozik said.

A look at upcoming guests reveals a mix of celebrities like George Clooney and Scarlett Johansson, politicians Jeb Bush and Vice President Joe Biden, and business leaders like Tesla founder Elon Musk.

\"And I think there may be a bit of a branding effort that, you know, fits with his actual persona of kinda being the thinking person's comedy show, you know,\" Poniewozik said.

And naturally there will be laughs. Colbert got his start on the improv stage of Chicago's Second City. It training that sharpened his wit and comedy agility.