Following the British-mediated peace in the War Between States and the shelved Emancipation Proclamation after McClellan's failure at Antietam, Lincoln's base with the Radical Republicans was demolished, and with his popularity declining in the wake of the Confederacy's creation, it was clear the Radicals needed a new voice within government. This voice was found in John C. Fremont. Not only a noted abolitionist, but one who had issued his own Emancipation Edict as Department of the West and had been promptly sacked. This, naturally became the cornerstone of his campaign. In the wake of Lincoln's apparent failure, Fremont was able to present himself as a noble man who had been restrained by the administration.
In spite of competition, the National Union Party persisted enough to nominate Lincoln again to serve a second term. The Democratic Party also found it's footing in General McClellan, although Democratic Party's position of reconciliation with the Confederate States was at the wrong place and wrong time.
While Fremont was able to be Elected, the Radical Democracy Party was short-lived as it soon found itself absorbed back into the Republican Party as the National Union Party finally collapsed. The Republican Party would launch a united Ticket next election under Ulysses S. Grant, however the division between the Radicals and the more Moderate wings (and later, the Stalwarts and Half-Breeds) would still cause one or the other to attempt to gain the presidency alone occasionally.
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