Obviously, slavery was a major issue among the states when Abraham Lincoln was inaugurated. Lincoln believed that he needed to communicate his desire to uphold the Constitution of the United States. He felt that southern states needed to understand this. For this reason, he states his responsibility as president to uphold the Constitution in the very first sentence.
In compliance with a custom as old as the Government itself, I appear before you to address you briefly and to take in your presence the oath prescribed by the Constitution of the United States to be taken by the President before he enters on the execution of this office."
He did not open his speech with this line by accident, he wanted to demonstrate that he was a faithful servant of Constitutional government early in the proceedings. The speech continues to express how Lincoln desires to respect the Constitution and his role within it. With regards to Article IV, Section 2 of the United States Constitution, (Fugitive Slave Clause), Lincoln directly addresses the issue of whether he would interfere with the institution of slavery. He starts by reading the direct text of the Fugitive Slave Clause.
There is much controversy about the delivering up of fugitives from service or labor. The clause I now read is as plainly written in the Constitution as any other of its provisions:
No person held to service or labor in one State, under the laws thereof, escaping into another, shall in consequence of any law or regulation therein be discharged from such service or labor, but shall be delivered up on claim of the party to whom such service or labor may be due.
It is scarcely questioned that this provision was intended by those who made it for the reclaiming of what we call fugitive slaves; and the intention of the lawgiver is the law. All members of Congress swear their support to the whole Constitution--to this provision as much as to any other. To the proposition, then, that slaves whose cases come within the terms of this clause "shall be delivered up" their oaths are unanimous. Now, if they would make the effort in good temper, could they not with nearly equal unanimity frame and pass a law by means of which to keep good that unanimous oath?
In speaking about this part of the Constitution, he not only discusses how the president was obligated to accept this law, but also the responsibility of Congress to as well. Lincoln felt that it was his responsibility to accept the laws of the land as he received them, and more importantly, had them over to the next president in the same condition. An important part of this during his presidency would be to respect the institution of slavery. In making his case that he would respect the law of the land, he is demonstrating an important responsibility that he has as president according to the Constitution.