What is the exact manner to join the Salat Jamat after arriving late?

To join the Salat Jamat after arriving late, there are specific steps to follow, as derived from the teachings of the Prophet Muhammad ﷺ and the Imams. One narration from Al-Kafi explains a situation where a person arrives at the mosque while the congregation is already in prayer. The Imam may face a problem and will signal for him to lead the prayer if he is close enough. The Imam said: He completes Salat (prayer) of the people, then sits down until they complete Tashahhud (testimonies of belief), he makes a gesture to his right and lifts and it (gesture) is considered saying Salam (the phrase of offering greeting of peace) and completion of their Salat (prayer) and he completes whatever he is yet to complete. Al-Kafi 57:5262.

This narration illustrates that if you arrive late and find the Imam in a rak‘at (unit of prayer), you should perform your takbir (the opening phrase of prayer) and join in the prayer. If you find that the Imam is in the final sitting position, you can join by performing the prayer in your place and completing your own prayer afterward.

The emphasis on following the Imam's actions and completing your prayer highlights the importance of unity in congregation while also allowing for individual completion of prayer.