The underground area starts on the small island right off the front of the main island, by the gate.

You'll see a long corridor made of a non-climbable surface with some lights up top that you can use to swing to the end of it.

At the end, the corridor branches off into two paths. Both sides are stand-alone and have no links to each other or any puzzles on the surface.

Left:

• Swing onto to ledge at the top of the far end of the left corridor, past the two energy-walls.
  
• Grapple onto something and lower yourself down the hole.
  
• Swing into the side-room and climb the wall.
  
• Use the button to disable the two energy-walls you saw on your way here.
  
• Climb your way to the top and solve the arrow puzzle.
  
• Make your way to the viewing chamber, collect your reward.
  
• Use the switches to disable the energy-walls and exit.

Right:

• The light source at the end is blocked by an energy-wall, you'll have to swing or glide over to it.
  
• While sliding down, look for a light source you can grapple onto and pull yourself over to it.
  
• Climb the pillars and make your way to the docks.
  
• Collect your rewards.
  
• Use the switches to disable the energy-walls and exit.