Shadow of the Colossus/Gaius - Colossus 3

Reaching Gaius
To reach the general area where this colossus is, go north across the natural bridge. Instead of going down into the valley like you did with Quadratus, keep going north. Find the path through the mountains to the left and it will lead to a lake.

The third colossus can be a bit tricky to reach, mostly because the jumps involved in reaching the canopy it's sleeping on are quite far. If you don't time your jumps correctly, you'll end up falling back into the lake and have to swim back to the ramp. If you make it onto the canopy, follow the path onto the surface where you'll be greeted with a cut-scene of this colossus, named Gaius the Knight, waking from hibernation.

Defeating Gaius
The first order of business is to find the small circular platform on this roof; it's vital to getting onto the colossus. Keep your distance as you run around searching for it, and if you've got your vision locked on to it, be careful not to accidently run off the edge. Once you've found the platform, wait there and watch Gaius. More than likely it'll start to position itself for attack; the obvious sign is that it will draw back the "sword." This is you cue to run; however, if you want to progress further in this battle, run directly away from the colossus rather than to either side. If you've done this correctly, its "sword" will strike the platform and knock off a portion of the armor on its arm.

Run back to the platform and wait for him to strike again. The difference you don't want him to actually hit the platform this time, so ignore what I said earlier and run off either left or right of the platform. Gaius will likely strike the ground with such force that its weapon is briefly submerged in the ground. Take this opportunity to jump and grab onto the edge of the sword until Gaius frees it from the earth. Next pull yourself onto the sword and run towards its arm, grab on, and climb all the way to the top of its head. This is where the first target is located. Charge strike until the symbol disappears.

The final weakpoint is located on the stomach. Climb down its back and drop down to the platform circling its waist. Moved to the front and latch on to the stomach. The colossus will violently try to shake you off (as usual), but hold on tight and charge strike the symbol. Remember, you can rest if needed by dropping back down on the waist-platforms, crouching so as not to fall if he tries to shake you off again. Eventually the big guy will go down and you'll be brought back to the temple.

Hard Mode Strategy
The final magic symbol on hard mode is located on the back of its left arm (the one not holding the "sword"). You can either attempt to jump across to it, or go back up to Gaius' shoulders and climb down using the fur. Find the glowing mark. Your stamina still isn't great and there is nowhere to rest, so you'll have to move fast. A few charged strikes later and Gaius is finally down for the count.

Just as a word of warning: if at any point Gaius makes a direct hit on you with his weapon, you are instantly killed.