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The general and probably accurate assumption is that Robert E. Lee becomes the President of the Confederacy after Jefferson Davis in Confederate Victory Tls. He much like Washington, for the young USA wins more or less unopposed and for a time at least keeps the internaldivisions in the young Country at bay.
It seems to me that the Confederacy lacked a similar unifying figure should Lee not serve and thus potential internal divisions would manifest sooner. I would think that it would be a contested elecation more than a coronation for Lee. Who then serves as Davis's successor? Stephens? Perhaps another General as the Army seemed to be more popular and likely would have received more credit for a victory. Although I cannot think of who that would be perhaps Longstreet? Would it depend on what the exact boarders of the Confederacy would be?
Assume that this is rather late victory scenario like a TL 191. Lee either died postwar but before the end of Davis's term or perhaps was assassinated at a similar time in ATL as Lincoln was in our own timeline for a bit of historical irony.