The center fielder has priority over every other fielder on all fly balls. Anything they think they can catch they should call for and make the play.
On balls hit to left or right field, the center fielder should go for the ball in case the other fielders do not make the play.
The center fielder can also yell “forward” or “back” on fly balls hit right at the other outfielders in case they are having trouble judging the fly ball.
The center fielder backs up second base on all throws to second from the infield. On throws to second to get a runner stealing from first the center fielder must be careful to not overrun the ball. Overrunning the throw can result in the runner scoring.
Regardless of where the ball is hit or thrown the center fielder has somewhere to go and something to do. They should never just stand and watch.