Talk to Buck outside then mine for a short history lesson. Now enter the mine and ride the elevator. Push the cart and move silently, as there's several enemies inside as well. Clear the enemies in the cave until you reach the first pirate encampment in the cavern.
There's a couple of snipers guarding the entrance so take cover behind the pushcarts and push them as you go forth. You can also take them out from a distance should you wish to do so. Next, eliminate the two guards across the bridge. Be careful of the two charges and a heavy in the area as well.
Continue forth to find another passage filled with pirates. Take them out then continue forth to the next door. Use the compass to open it. Head further inside until you reach a dead end. Activate the circle in the middle to reveal a passage. Check out the opened wall panels around for some additional loot as well.
Continue outside and follow the path until you reach a wooden elevator. After barely escaping from being crushed, continue further until you reach the temple. Once inside, loot the boxes and grab the third part of the compass. Sprint ahead as the cave around you starts to crumble. Continue until you reach the exit.