Changing one''s intention from

A: Yes, it is permissible for the latecomer to follow the lead of the person who joined the congregational Salah for part of it and then stood up to perform the missed Rak`ahs after the Imam had said the Taslim (salutation of peace ending the Prayer). The basic principle is shown in what was narrated by Abu Dawud and Al-Tirmidhy, and was classed as Hadith Hasan (a Hadith whose chain of narration contains a narrator with weak exactitude, but is free from eccentricity or blemish) by Ibn Khuzaymah and as Hadith Sahih (a Hadith that has been transmitted by people known for their uprightness and exactitude; free from eccentricity and blemish) by Ibn Hibban and Al-Hakim: The Prophet (peace be upon him) saw a man praying alone. He said, "Is there a man who can be charitable to this (man) and perform Salah with him?" The Six Hadith Compilers (Al-Bukhari, Muslim, Abu Dawud, Al-Tirmidhy, Al-Nasa'y, and Ibn Majah) narrated on the authority of Ibn `Abbas (may Allah be pleased with him) that he said, I stayed overnight in the house of my aunt Maymunah. The Prophet (peace be upon him) got up to perform Qiyam-ul-Layl (optional Prayer at night) and I got up to perform the Salah with him and stood to his left. He held my head and drew me to his right. Also, Ahmad and Muslim narrated on the authority of Anas (may Allah be pleased with him) that he said, (Part No. 7; Page No. 405)  While the Messenger of Allah (peace be upon him) was performing Salah during Ramadan, I came and stood behind him. Then another man came and stood by my side, then another one came until we had become a group. When the Messenger of Allah (peace be upon him) perceived that we were behind him, he lightened the Salah. He then went to his house and performed a Salah, (the like of which) he never performed with us. When it was morning, we asked him, "O Messenger of Allah! Did you notice us in the night?" He said, "Yes, it was this that made me do what I did." `Aishah (may Allah be pleased with her) narrated that The Messenger of Allah (peace be upon him) used to offer Salah in his room. As the wall of the room was low, the people saw Messenger of Allah (peace be upon him) and stood up to follow him in Salah, and they spread the news. The following night the Messenger of Allah (peace be upon him) got up to perform the Salah and the people followed his Salah. (Related by Al-Bukhari) These proofs affirm the permissibility of taking someone who is performing Salah alone as an Imam during a Nafilah (supererogatory) Salah. The general ruling is that there is no difference between the Fard (obligatory) and the Nafilah (supererogatory) Salah in this, unless there is evidence that indicates a specification of that general principle. Accordingly, being a latecomer does not prevent others from taking you as an Imam when you are performing Rak`ahs that you missed to attain the reward of the congregational Salah. This is the most correct opinion of the scholars.May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family and his Companions!