Republicans like to boast that they're the party of Lincoln, but they're not.
In Lincoln's time (1800s), Democrats were conservative and Republicans were progressive. But the 1964 Civil Rights Act (prohibiting people from discriminating based on race or gender) caused a ton of white conservative Southern Democrats to switch to Republican. This, in turn, caused a ton of liberal Republicans to go Democrat, and our parties essentially switched ideologies.
So if Lincoln were alive today, he'd be a Democrat, not a Republican.