Mention of the disagreement over the caliphate, then the consensus on the caliphate of Abu Bakr (may Allah be pleased with him)

It is narrated from Sayyiduna Abu Sa'id al-Khudri (may Allah be pleased with him) that when the Messenger of Allah (peace and blessings be upon him) passed away, the orators of the Ansar stood up, and a man among them began saying: "O company of the Muhajirun, whenever the Messenger of Allah (peace and blessings be upon him) appointed one of you as an official, he would pair one of our men with him. So our view is that this matter, that is the caliphate, should be taken charge of by two men, one from you and one from us." The narrator said: The orators of the Ansar then kept to this same view one after another. Then Sayyiduna Zayd bin Thabit (may Allah be pleased with him) stood up and said: "Indeed the Messenger of Allah (peace and blessings be upon him) was from the Muhajirun, and the Imam will also be from the Muhajirun, and we are his Ansar, that is his helpers, just as we were the Ansar of the Messenger of Allah (peace and blessings be upon him)." Then Sayyiduna Abu Bakr (may Allah be pleased with him) stood up and said: "O company of the Ansar, may Allah reward you with good and keep your speaker firm upon the truth." Then he said: "Had you done anything other than this, we would not have made peace with you." Then Sayyiduna Zayd bin Thabit (may Allah be pleased with him) took hold of the hand of Sayyiduna Abu Bakr (may Allah be pleased with him) and said: "This is your man, so give him the pledge of allegiance." Then everyone left. When Sayyiduna Abu Bakr (may Allah be pleased with him) sat on the pulpit, he looked at the faces of the people but did not see Sayyiduna Ali (may Allah be pleased with him) there, so he asked about him. Some men of the Ansar went and brought him. Sayyiduna Abu Bakr (may Allah be pleased with him) said: "O cousin of the Messenger of Allah (peace and blessings be upon him) and his son-in-law, did you intend to break the unity of the Muslims?!" He replied: "No blame, O Successor of the Messenger of Allah." Then he gave him the pledge of allegiance. Then he did not see Sayyiduna al-Zubayr bin al-Awwam (may Allah be pleased with him), so he asked about him until the people brought him. He said: "O son of the paternal aunt of the Messenger of Allah (peace and blessings be upon him) and his disciple, did you intend to break the unity of the Muslims?!" He too gave the same reply: "No blame, O Successor of the Messenger of Allah." So both of them gave the pledge of allegiance.
This hadith is sahih (authentic) on the condition of Imam Muslim, but the two Shaykhs (Bukhari and Muslim) did not record it.

It is narrated from Sayyidah Aishah (may Allah be pleased with her) that when the Prophet (peace and blessings be upon him) was taken on the Night Journey to al-Masjid al-Aqsa, in the morning the people began talking about it. Thereupon some of those who had believed in him (peace and blessings be upon him) and affirmed him (peace and blessings be upon him) apostatized. Some of the polytheists hurried to Sayyiduna Abu Bakr (may Allah be pleased with him) and said: "Have you heard anything about your companion? He claims that last night he was taken on a journey to Bayt al-Maqdis!" Sayyiduna Abu Bakr (may Allah be pleased with him) asked: "Did he really say that?" They said: "Yes." Sayyiduna Abu Bakr (may Allah be pleased with him) said: "If he said that, then he has certainly spoken the absolute truth." They asked: "Do you even affirm that he went to Bayt al-Maqdis last night and came back before morning?!" He said: "Yes. I affirm him in something even more far-fetched than that: I affirm him without hesitation in the news that comes from the heavens, that is revelation, morning and evening." For this reason he came to be called Abu Bakr al-Siddiq.
This hadith is sahih (authentic) on the condition of the two Shaykhs (Bukhari and Muslim), but they did not record it. Indeed, Muhammad bin Kathir al-San'ani is saduq (truthful).