After talking to Duncan you learn he thinks there are thirsty cats inside the Establishment and he wants to sell them milk.
Duncan will leave if you remove the sign. You can access the sign from the rooftop.
This is the low-spoiler guide for "Remove the sign" puzzle in Nine Noir Lives. Unlike in normal walkthrough, you get gradual hints on how to solve the puzzle. Click the big button below to see your first hint. If the hint contains a spoiler, you will be warned about it before clicking.