When Douglas accuses Lincoln of favoring black citizenship, he makes it very clear that he is against it. What instead does Lincoln favor ? How does he propose to achieve his goal?
Abraham Lincoln was first and foremost a politician on this issue, which can make him difficult to pigeonhole due to varying degrees of the issue, be it slavery, equal rights for blacks, and/or black citizenship. Despite the modern understanding of a politician, Lincoln was more akin to a pragmatist on this issue, and was know to speak in favorable terms to his audience, but without committing to a definite view, as the best kind of moderation calls for.
Lincoln favors citizenship rights for blacks based on reluctant enforcement of the Fugitive Slave Act. By doing so, during the 1850's and especially as president, he was able to reconcile federal law with natural law and beliefs, as a way to call attention to his disagreement with the Dred Scott case without explicitly calling into question the legitimacy of the Supreme Court. This proposition was essentially to work on a case by case basis as a kind of persuasion. Before the outbreak of the Civil War, Lincoln likely hoped that by pointing out the inconsistencies of slave regulations throughout the country, that a majority would be persuaded through relative understanding. This was a masterful stroke to wading through the morass that was the "peculiar institution" of slavery, especially as war over slavery was not seen as inevitable until its unfortunate outbreak following the firing on Fort Sumter.
His legacy (as cut short as it was) was ultimately to defer to the states as the non-egalitarian questions of black citizenship versus the larger debate on the national level, hoping that fairness would eventually overturn discrimination based on legality. While this wouldn't happen until the Civil Rights Era, it wasn't the incorrect decision.
Perhaps more than any one person in the history of the United States, it is essential to examine the actions and attitudes of Lincoln's time before judging his somewhat unsatisfactory (in a modern context) solutions to the question of racial equality. A lot of things in history ultimately work like this, as it can be said that civilizations in 500 years would likely judge our current world harshly when it comes to income inequality and other issues if the using the same flawed relative perspectives.