Is the prayer of one who caught the Imam during the Eid prayer while he was reciting Al-Fatiha considered valid, and is there nothing upon him?

If someone misses the additional takbirat with the Imam in the Eid prayers and joins him during the recitation, they should say the takbir for ihram (definition) and then perform the additional takbirat. This is the view of the Hanafis and Malikis, and the old view of the Shafi'is. As for the new view of the Shafi'is and the Hanbalis, they said that if one misses them, they do not make up anything, because it is a recommended remembrance that has passed its place. The preponderant opinion is the first one, which is the recommendation to make them up. Your prayer is valid, because the takbirat and the dhikr between them are sunnah (definition), and the prayer is not invalidated by intentionally or forgetfully omitting them.