I think that Lincoln was able to show his understanding of statesmanship and what leadership represents in his attempt to address the Southerners. Lincoln recognized that the Cooper Union speech would be seen as his belief of Republican identity. He also understood that his desire for national leadership meant that he must address national issues to the nation, at large. The second section of the speech is carved out with this as a purpose. Consider Lincoln's opening of this speech as his own answer to the question of slavery and the question of how he appropriates the Southern mindset as a candidate for President:
And now, if they would listen - as I suppose they will not - I would address a few words to the Southern people.
In addressing as many of the Southern criticisms of both Lincoln and Northerners, in general, Lincoln shows that he recognizes what discordant parts of the population are saying about he and his beliefs. Lincoln would certainly like to have won some Southern hearts and minds in his speech, but he understands that it is more important to firmly carve out his political position in answering Southern claims against his position and, by extension, the Republican stand. Lincoln recognizes that his fundamental premise is adherence to the Constitution and not deviating it from it in terms of preservation of the Union and forbidding slavery. Lincoln attempts to present the Republican case in addressing Southern criticisms. Yet, he is also firmly drawing a line that he wants the South to understand. In contrast to Douglas, who presented expedient and political solutions to the issue at hand, Lincoln is firm enough in both his convictions and his political base to understand that he must state without hesitation that Southern demands are not going to guide national policies. Lincoln is both calling out to the South, but his real purpose is to call out to the Republican base in shoring their support for him. Lincoln understands that the political demographics are skewed enough for him to win the Presidency if he can clearly represent the Republican position and embody the Republican candidate. It is in this light that he speaks to both Southerners, but more so to his own party in making sure they support his candidacy. His message to the Southerners is more of a warning that he will not retreat from the positions established in the speech. It is a reminder that Lincoln's greatness was his political courage at a time when others demonstrated it as a resource in short supply.