I am asking this because I am not a CFI and don't know the answer and not trying to be a smart ars.

How do CFIs log time? For example you have a student with a private that signs for the plane at the FBO and is the person at the controls. The instructor is say.... doing a BFR or refresher training or something. Something that wouldn't require them to be in the aircraft and the student could be flying alone.

I know the student would be the PIC because they signed for the A/C and also are the one at the controls. What does the CFI log? Is dual given a form of PIC when you are a CFI?