[Ruling on someone beginning his prayer alone, then another joins him]

Answer: If the first man intends imamate when the second joins him, the prayer is valid in congregation. But if the first man does not intend imamate at that point, then the prayer of the second man who intended to follow is not valid. This is what is stated in the marginal notes on Sharḥ al-Azhar according to the school.
I say: This can be supported by what is narrated of the Prophet’s approval—may Allah bless him and his family and grant them peace—of Ibn ʿAbbās and Anas b. Mālik when they came and prayed along with him while he was in prayer; and by analogy to the substitute imām (al-mustakhlaf), who intends imamate after having been a follower, and his intention is valid and the prayer behind him is valid.
Source : Min Thimār al-ʿIlm wa al-Ḥikmah vol.1