That which was a matter of dispute and disagreement after...
That which was a matter of dispute and disagreement after the death of the Prophet was the question of succession to the political leadership, not the Imamate, which included the spiritual dimension of the Prophet’s legacy. No one at the Saqifah had anything to say about choosing an Imam, and the question was not even raised.
This is because nobody had the least doubt concerning Imam Ali’s supremacy in spiritual matters, or was it because none of the claimants to the caliphate and the succession was qualified for the Imamate no one laid claim to The truth of the matter is unclear. For some time then there was no mention of the Imamate.
But, after the death of several of the caliphs, the question gradually came to the fore, and some of the caliphs, like Mu’awiyah, for all his lack of commitment to Islam, began calling themselves Imams. The topic discussed in works of theology is the Imam and the Imamate, while the terms used in books of history and the oral and written statements of Sunni scholars are caliph and caliphate.
Imam Ali and his descendants, recognized as the leaders of the Shi’ah are however consistently referred to as Imams. This reflects the Shi’i belief that strict and precise adherence to the criteria of religion, unswerving piety, and a whole series of other special qualities, must be present in the person of the Imam. One of the pupils of Imam Ja’far al-Sadiq, (PBUH), Hisham b. Hakam wrote a book on the subject of the Imamate in which he set forth its theoretical bases.
(9) In addition to the office of prophethood, which comprised the responsibility for receiving and conveying revelation to mankind, the Prophet was the ruler of the Muslims, empowered over all their affairs. From the moment that the Muslims established a collective existence; all the social affairs of the people were regulated by the Prophet: the appointment of governors, commanders, and judges; the distribution of booty; the issuance of orders for war and so on.
He implemented divine commands and ordinances in accordance with the ruling function that was vested in him, and it was the duty of the people to obey his commands and instructions. Rulership, the administration of society, and the establishment of public order and security were thus part of his prophetic function; prophethood and spiritual leadership on the one hand and leadership and rule on the other were both combined in a single divinely chosen person.