In his last days, knowing death was imminent, Prophet Muhammad called his uncle Abbas and Imam Ali. He asked his uncle to carry out his will and settle his affairs after the prophet's death. When the uncle demured, the prophet asked Imam Ali, who accepted. The Prophet also gave Imam Ali his sword, battle gear, and ring.
Three days before his passing, the prophet went to the masjid and asked if anyone had a grievance against him, and offered to let them seek retribution. A man stepped forth saying that the prophet had accidentally hit him on the stomach once. The prophet said he could seek whatever retribution eh wanted. The man asked the prophet to remove his shirt and then kissed the prophet on his stomach.
At one point Ammar bin Yasir asked the Prophet how his ghusul and burial should be conducted, and the propeht replied that it would be done by "Ali ibn Abi Talib, for no part of my body will he touch without the angels aiding him in doing so"
Another time, when the prophet was weak, he woke up from sleep and asked for "my brother and companion." Aisha said to summon Abu Bakr, when when the prophet woke up again saw him sitting there he turned away from the man. After Abu Bakr's departure the prophet again asked for "my brother and companion." Usman's daughter Umm Salma had Usman summoned, yet he received the same treatment fromt the prophet. When the prophet called for "my brother and companion" a third time, Imam Ali was summoned and the Prophet spoke to him privately for a long time.
When asked what they spoke about, Imam Ali replied "He taught me a thousand doors (of knowledge) and each door opens a thousand more. He also entrusted me with responsibilities that, God willing, I will fulfil"
Lecture notes available at http://www.why-quran.org/?p=800.
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