I should have clarified by just saying Lawful. Lawful characters take the oath seriously. Mookie indicated that he could retain his honor by challenging Ulfric after the conclusion of the civil war questline. I countered with the oath, showing that he was bound by it to recognize not only Ulfric as his lord and High King, but to do so for life. Now, any alignment can swear an oath, but a lawful character, who would be concerned about his honor, cannot violate it. As such, challenging Ulfric is a dishonorable act. While Neutral and Chaotic characters would not care about this loss of honor, a Lawful character would be. As it is heavily implied that Mookie is playing such a character, I pointed it out to him.
EDIT: As you can see in post 95, he even uses the term Lawful.