There should be no way it is leaking from the cap unless the cap isn't on correctly.
The cap should be carefully placed on the radiator where the tabs line up, and then turned clockwise a 1/4 of a turn until it stops. A stopper and bolt should be present to keep the cap from turning itself off.
2007 FZ6 - FZS6W Yamaha Motorcycle RADIATOR HOSE Diagram and Parts Part: #22
All overflow I believe should be coming out of the reserve tank behind the headers. (Or under the tank if you have a 2004-2006 FZ6)
I would keep an eye on the coolant temperature level that is indicated on your dash in the top right corner. At 200°-210°F the cooling fan behind the radiator should kick on to cool the coolant. Check to make sure it does, because if not, either the fan or fuse needs replaced. Check the fuse first since it is easier. (Right pod under the seat)
If you have a 2007-2009 FZ6, your cooling fan may have gotten burnt out due to a rock being jammed in the fan blades. That is what happened to me and many other S2(2007-2009) owners. A manufacturing defect on the S2 Fazer prevented the fan fuse from being blown before the fan motor went out due to being jammed. This coupled with the different fan design from the S1 meant that a rock could get perfectly jammed between the fan blade and one of the mounting arms on the fan, thus burning out the fan.
A fan is $80 from a scrapyard, I opted for a $20 automotive fan from Amazon instead. If you do end up having to replace the fan, I would look into making (or buying from forum member TownsendsFJR1300), a little strip of metal that mounts to the bottom of the radiator that prevents rocks from getting lodged in the fan. I would bet a dead fan is the reason your bike is overheating. When was the last time you flushed coolant though?
There are many threads on the dead fan syndrome on here, search in the search bar if you want to read them.
Welcome to the forum! :welcome: