Once you climb onto the top of the crumbled entrance of the Coliseum, swan dive directly into the water.

When you emerge in the first warehouse, be sure to kill all the enemies there (two rats and a couple of bats), so they don't bother you on your way through this room.

As you see in the screenshot to the left, jump to the ledge on the wall to the right, and from there to the ledge opposite you. That route is a tad faster than it would be if you decided to jump over to the pole on your left.

Swan dive over this board, so you continue on your way smoothly.

When you reach this point, jump on the slope directly from the stairs, and, as you land, jump immediately to land on the ledge ahead.

When you pull the lever and release the two gorillas, drop down from the ledge where you came, onto the ground. That will trigger the gorillas to emerge from their hideout. Quickly climb on the ledge again (using the rock on the floor) and shoot them from safety. Repeat the same process for the remaining two lions.

Once in the second warehouse, grapple the cage directly from that small ramp you're standing on, and pull it all the way towards you.

The next sequence is shown in the screenshots to the left. Grapple the ring above you and swing to the other side. Then turn left and jump to grab the crack on the column. Lara will probably start to slip, so save her before she falls. Traverse to the right, and jump behind you to another ledge. Traverse right again, and jump up to grab the upper ledge. From there, continue on your way.