1- In this letter, Imam Ali (a) is refuting Mu’awiya’s claims and leaving no excuses for him. First of all, Mu’awiya recognizes the caliphate of those 3, so the Imam is telling him what’s different about me? If they were legitimate rulers them so am I, so on what basis do you fight me and reject my authority?
2- Secondly, Imam Ali (a) indicates that the Muhajirin and Ansaar unanimously pledged allegiance to him, and Allah is pleased with the person who unanimously was chosen. Now on the outer surface this statement could be seen as supporting Abu Bakr, but in reality it isn’t because there was no unanimous allegiance to Abu Bakr. Imam Ali refused to pledge him allegiance, along with a few others. So the statement of the Imam that the one who is unanimously chosen is accepted by Allah does not apply to Abu Bakr.
So the point of this letter was just to refute Mu’awiya even based on his own standards:
1- He believed those 3 were legitimate, so then why isn’t Imam Ali’s caliphate legitimate? What’s the difference?
2- Mu’awiya may try to argue that Muslims unanimously chose Abu Bakr (which isn’t the case of course), and so the Imam is telling him: Muslims of Medina unanimously chose me (so why don’t you obey me?)