Yes, that's correct. Key 3802 opens both Dwacatra's Prison and the Mine Hub in the Dwarf Mines. It is obtained during the Emperor's Cookies Quest.
Start by entering Emperor Kruzak's private chambers, located north of the throne room. Inside the chamber, take the hidden westward passage. Follow it to the end to find a quest chest containing Key 3800.
Do not go down the hole at the end of this passage. Instead, backtrack to the emperor’s chamber. With Key 3800, unlock the small side room inside the chamber. The chest in this room contains Key 3801 and cookies.
Now take Key 3801 and head to the barracks (first floor of Kazordoon). There is a locked room here that opens with Key 3801. Inside that room is a quest chest containing Key 3802.
Key 3802 can now be used to access Dwacatra's Prison and the Mine Hub.
If you prefer visual guidance, the TibiaWiki page for the Emperor's Cookies Quest includes a detailed map for the exact paths and room locations.