Sheet music is not merely a system for indicating where to press the keys. It functions as a message card that communicates what is happening musically. It might be helpful to compare it side by side with the key of C.
In solfa terms, the quality change from IIIm to III involves so turning into so♯, meaning the fifth note of the key is raised by a semitone. In the key of A, this corresponds to E becoming E♯. Calling it F would imply that the note originally was F♯ and has been lowered to F, which doesn’t accurately represent what happens during the transition from IIIm to III in this context.
