double arrow sign at "T" intersection...what are the rules?

[daveski]

There is an intersection near me that I travel every day and it is at a T .
there are 2 lanes of traffic at the "T" ....one presumably goes right, one goes left.
I turn left to get to work.
Occasionally the car to my right will make a "wide left" turn.
Exactly what are the rules that apply to when you get to a "T " intersection and you see the double arrow sign in New Jersey?