After the scenes, you will be thrown immediately into a battle against two PSICOM Raiders and an Infiltrator. The Infiltrator is stronger but the Raiders can inflict status ailments, so they should be removed first before focusing on other enemies.
After the battle, simply approach the waypoint and have Lightning jump her way to the next area. Flip the switch to activate an elevator and once at the top, jump up some more to the next path. When you finally reach the long corridor, turn south and collect the 4x Digital Circuit from the Sphere at the end. Follow the corridors round to the waypoint for a further scene.