From the Super Mario Wiki
The Second-Degree Card is an item Mario receives in Paper Mario after defeating Lee in the Toad Town Dojo, who is available to fight once Chan is defeated and the First-Degree Card is obtained. This item proves his strength and allows Mario to fight The Master for the first of three times. Additionally, there is an unnamed yellow Dryite who has a Second-Degree Card as well, meaning that the Dryite fought Lee to gain the Second-Degree Card while visiting Toad Town. The card has no particular use in-game other than affecting dialogue at times, and is simply a collectible item.
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