Islamic History

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Sister, your question has a saliant presumption  and Khala Ummi Talib's responses don’t answer this anomaly. In my humble view this presumption needs to be quashed.

1. Factually and historically, Sayyida Aisha (RA) NEVER OBJCTED to the caliphate of Sayyidina Ali (RA) at the time (of Sayyidina Ali (RA)) at assuming caliphate. There is no evidence to suggest otherwise at all from either Shia or Sunni sources.

Having established this fact, now read Khala Ummi Talib's summary about the battle (iteself) or a very detailed response here. 

Masha'Allah, when that is clarified lets move to the actual issue, itself.

This is one of the most Authentic Accounts between the two, ponder over the wordings and the exchange, I am going to break the conversation into three easy chunks:

وروىابن حبان أن عائشة – رضي الله عنها – كتبت إلى أبى موسى الأشعري – والى علىّ علىالكوفة-: فإنه قد كان من قتل عثمان ما قد علمت، وقد خرجت مصلحة بين الناس

Ibn Habban narrated in al-Thiqat that Sayyida ‘Aisha (RA) wrote to Sayyidina Abu Musa al-Ash’ari (RA) and to Sayyidina ‘Ali (RA) in al-Kufa: “You have witnessed the events leading to the murder of ‘Uthman and I have decided to go and fix the matters between the people…”
 

وقال:أي أُمه، ما أشخصك وما أقدمك هذه البلدة؟ قالت: أي بنى، إصلاح بين الناس

Sayyidina ‘Ali (RA) then sent al-Qa’qa’ ibn ‘Amr al-Tamimi (RA) to Sayyida ‘Aisha (RA) and he said: “O mother, what brings you to these lands?” she said: “my son I have come to settle matters between the people.”

And after the war of al-Jamal Sayyidina ‘Ali (RA) told Sayyida ‘Aisha (RA):
 

فقاللها: غفر الله لك. قالت: ولك، ما أردت إلا الإصلاح

“May Allah forgive you” she (RA) said: “And you, I only wished to fix the affairs.”

1. Two people calling each other Mother and son
2. Making dua for each other even after the battle

Does it sound like a nasty battle like the Shias portray? No, sounds more like a mis-understanding to me!