NFHS rules.
1st inning catcher #14 gets a hit. #3 comes in as a courtesy runner for catcher #14. 4th inning coach changes pitcher and catcher. The catcher #14 is now the pitcher. 5th inning pitcher #14 get another hit. Can # 3 be her courtsey runner again.

#3 is a CR2; a courtesy runner for catcher. She is not a designated courtesy runner for #14. Answer is no.

CR is for position and not player. Thus #3 is CR for F2 position and not #14 whom
has changed position.

My understanding is that a CR "runs" for either the pitcher OR catcher, not both. I agree that CR is designated to either F1 of F2 and not an individual player.

I agree with Glen. The designation is by position not player.