It all depends on what obj.method() does with it. If it calls it immediately, then just call the method that calls obj.method(). If it saves it for later, then make sure the later operation is executed by your test.
I would tend to turn non-trivial lambda functions into methods and simply use a method reference. This way you keep the advantage that the lambda affords - succinct code to pass callbacks - but also keep the advantage that a method affords - visibility as separate units.
Wouldn't the advice to not test private methods apply here? Test the behavior that you expect from the public method which uses this lambda, and you'll save yourself headache if you refactor later.
obj.method()does with it. If it calls it immediately, then just call the method that callsobj.method(). If it saves it for later, then make sure the later operation is executed by your test.