The short answer is no, if it's his time & he's not available, then he'll need to provide for child care for that time. Communicate same to him on a timely basis. You may also consult a NYC Child Custody attorney for a full assessment.
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No, you are not required to do that according to your agreement. Your ex husband should try to make some arrangement for the hours he won't be available. He is basically responsible for this. You, of course, want to be sure that his arrangements are good for your child, but it is up to him to make them. Speak with your divorce attorney.
He can hire a daycare person like any other parent.
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