After the briefing, enter the hotel and approach Muggshot. A cutscene then plays, before a thought bubble appears above Bentley's head, which informs you that you need to insult Muggshot to provoke a fight. The thought bubble will give you three options, mapped to , and . None of the default options will be the correct response, so choose any option. This will happen two more times before the correct option, insulting his mother, will appear as the option. After insulting his mother, Muggshot accepts the challenge to a fight in Town Square. Control will switch to Sly, who is out in the field.
After receiving instructions, track down Carmelita, get her attention, and take the chase to Town Square. When you get there, a cutscene plays where Sly escapes, Muggshot arrives and Muggshot and Carmelita face off. Control will switch to Carmelita.
You'll then have to face Muggshot. Muggshot will know exactly where you are the whole time. It is recommended that you stay out of his direct line of sight. If you're in his sights but not right next to him, he will start shooting and only stop if you hide behind something or get enough distance between you and him. If you decide to get next to him to attack, he will start kicking you. You also should not kick Muggshot; your kicks will only do minimal damage, while Muggshot's kicks will take out a good portion of your health. It's best to stick to the rooftops and shoot Muggshot from there. This makes it easier for rooftop guards to attack you during the fight (you can handle them fairly easily), but this also means that Muggshot will need to resort to throwing explosives near your position. Get out of the way of the explosives while continuing to shoot at Muggshot. Once he's down, it's Job Complete.