Actually, the "radical" abolitionists such as Garrison and Phillips were opposed to war and did not support Lincoln (at least in the early years of the war.) They had argued for years in favor of "Free State" secession from the Slave States, and had no use for the Constitution which they saw as a compact with evil since it acknowledged slavery to be legal. (Garrison actually burned it in public as a protest.)
The "one issue" people, radical abolitionists in the North and Fire Eaters in the South, just like today's Cindy Sheean and her Code Pink buddies, didn't give a damn about Union. They wanted it "their way" and they wanted it now and to hell with everyone else!
They were also united in their hatred, disdain for Abe.