It is not a condition for supererogatory fasting to make the intention before dawn. This is proved through the Prophet’s practice. `A’ishah (may Allah be pleased with her) stated that the Prophet (peace and blessings of Allah be upon him) entered upon her one day at forenoon time and said, “Do you have anything (to eat)?” She said, “No.” He said, “Then I am fasting.” (Muslim)

It means that the Prophet intended fasting during day time. But scholars said that a person should not have eaten anything from dawn until he intends the fasting. They further said that it can only done as long as Zhuhr time has not entered.

Allah Almighty knows best.