Just like any ground ball, it only matters where the ball is picked up. If the ball hits the catcher and then bounces into fair territory, it is a foul ball, assuming the catcher is still in foul territory.

The ball is foul! The batters and catchers boxes are both in foul territory, therefor it is just like a ball hit down the first base line and lands in foul territory, it is foul. On the other hand, If a ball hits 1st or 3rd base on a fly, the ball is fair.

If the catcher and ball are both in fair territory and the ball hasn't touched a foreign object then the ball is fair.

This question is fundamentally the same (i.e. has the same answer for the exact same reason) as the similar question, "If a ball is batted down the 3rd base foul line, hitting the ground first in foul territory but rolling into fair territory (before going past the base) where it is then touched by a defender, is it a fair or foul ball?" Of course, it is a fair ball. The position of the fielder's body or any portion thereof is completely irrelevant and is considered superfluous information. All that matters is the position of the *ball* relative to the foul line at the instant it is touched by a fielder. Whether the ball first traversed into foul territory is irrelevant; in fact it can cross over from foul to fair or vice versa an unlimited number of times (before going past 1st or 3rd, that is). Also, to hopefully clear up what seems to be another related misunderstanding, the 'foul' lines are defined to be straight lines extending from the apex of home plate (i.e. the point at the rear of the plate) continuing straight down the 1st, 3rd base lines onto and over the foul pole. This means part of the batter's box is indeed in *fair* territory (some incorrectly believe the batter's box is entirely in foul territory). It also means, by definition, that not only are the entire portions of 1st and 3rd bases in fair territory, but the entire *plate* is also fair territory.