Actually your type of question comes up regularly
You can write or log what you like in your own log book. Its what you are counting for the grant/renewal of a licence or rating that is important.
Only one pilot can be PIC, unless the PIC becomes incapacitated in flight.
Ensure you agree in writing (Technical Log/ Authorisation sheet) before flight who is PIC, lack of clarity can come back to haunt you if there is an incident.