What is the ruling on an individual's prayer if a congregation joins him, and is he obliged to pray aloud and complete it as their imam, or not?

The transition of a solitary worshiper to the intention of leading the prayer (Imamah) is permissible according to many scholars. This is the view of the Shafi'i and Maliki schools, who did not stipulate the Imam's intention for Imamah. However, they recommend that the Imam intend Imamah to avoid disagreement and to attain the virtue of congregational prayer.

Sheikh Al-Uthaymeen preferred the permissibility of a solitary worshiper transitioning to an Imam in both obligatory and supererogatory prayers, because what is established for supererogatory prayers is also established for obligatory prayers unless there is evidence to the contrary. He based his argument on the Hadith of Ibn Abbas with the Prophet, peace and blessings be upon him, regarding the night prayer, where the Prophet, peace and blessings be upon him, transitioned from praying alone to leading the prayer.

Therefore, your transition to the intention of leading the prayer was permissible; in fact, it was preferable to attain the virtue of congregational prayer. It is important to note that reciting aloud in the Fajr prayer and the first two units of Maghrib and Isha prayers is a Sunnah for both the solitary worshiper and the Imam.