Personally, I'd let them know that for the least chance of confusion and error that you need days to be designated either mother days or father days. Whoever is there when you begin care of the child in the morning is the only one you will meet or communicate with that day and the only one you will release the child to that evening.
If that is not workable for them, I'd ask for a written schedule, weekly with who has the child in the morning and the evening, and if there are any midday "visits" I'd insist they both sign off on the schedule.
So sorry you are in the middle of this.