What is that part for? Anyone knows?

Gearbox cooler IMO!

Is gearbox under the seat?

It's the diesel fuel cooler.

I'm curious, why does the car need a fuel cooler??

Surely warmer fuel is easier to atomise and also closer to it's flashpoint?? Thats why some aircraft engines have fuel heaters on them.

Or is the cooler on the return line to the tank??

As lummox said in another thread, the cooler cools the return to stop it melting the plastic fuel tank

Aaaah! I see....... Kinda important then. Nobody wants a melted fuel tank really eh!!!!!!