For anyone who has not seen this done before, the O/P has a system like this installed.
It looks like soffit, but it's not.
I would have installed a piece of Pressure treated wood from the top of that double up beam that was sized to fit between the two 2X's
Then would have flashed the top of that beam with one piece of coil stock in the shape of a U up side down installed over Storm and Ice Shield to isolate it from the pressure treated wood. The U shaped piece would also have had two "tales" on the bottom to help devert the water to the drain system.
Caulking was just a very bad idea and will never work.
Also I have not used nails in any deck boards for at least 5 years, why use them instead of screws when nails are just going to come loose once the wood drys out. I even bought screw guns that I can use standing up.
With the newer ACQ wood I've seen to many cases where the so called ACQ nails still rust at the heads, so I use Ceramic coated screws.