Is it permissible to count a missed fast (qada') that was begun with a voluntary (nafl) fast intention from the night, and then to specify the intention for the qada' during the day?

If the fast is of different types, such as a missed obligatory fast (qada') and a vowed fast (nadhr), then it is necessary to specify the intention and to make it before dawn (tabyit an-niyyah qabl al-fajr), because no specific time is designated for it. A general intention of fasting is not sufficient. Therefore, what the questioner did is not considered a valid fast for either the missed obligatory fast or the vowed fast.