After Indiana recovers the Cross of Coronado on his scouting trip, he runs home to show his dad what he found. His dad ignores him. Doesn't even look up from his diary. Nothing.

Thanks, Pop.

Then the sheriff shows up with the bandits Indiana just outran and forces him to hand over the Cross of Coronado. The sheriff promptly gives the cross to the bandit in charge, a nameless robber wearing a fedora who, only moments before, was downright impressed by Indiana's ability to outsmart him on the train.

Everybody strides back out of Casa Jones (while Henry continues to ignore the entire affair), but just before he leaves, the man in the fedora turns to Indiana:

FEDORA: You lost today, kid, but that doesn't mean you have to like it.

Then he puts his fedora on Indiana's bowed head. When Indiana raises his head, we're greeted by the familiar face of the adult Indiana Jones, who smiles and then promptly gets punched in the face. Seems about right.

As a symbol, the fedora works in two ways. First, using it to carry out a time jump like this emphasizes the hat's transformative power. When mild-mannered archaeology professor Dr. Henry Jones, Jr., puts that hat on, he becomes Indiana Jones: fearless, charming, a world-class explorer. It's a source of power and identity for him. It's who he would be 24/7 if he didn't need that teaching career to pay the bills.

The fedora also symbolizes Indiana's quest for a father figure. Sure, he had a father, but Henry wasn't exactly a doting dad. When the sheriff shows up in his living room, Henry can't even be bothered to leave his study or ground Indiana until he's 40.

Meanwhile, the bandit who gave Indiana the fedora and a small morsel of encouragement clearly made a huge impression on him, given that grown-up Indiana's still wearing the hat today. The fedora represents the man young Indiana wanted to become from that day forward, and the man adult Indiana successfully became: a roguish adventurer who, unlike his AWOL dad, spends more time in the field than in his study.