The connector sentence is simply a way of making the wording flow better when you take one argument and use it to support another. Therefore, it must have the same meaning as the argument's title. If the meaning is different, it will appear to the reader as if the wrong support was added or the writer didn't realize what the argument was about. Or it might appear that a popular argument was misused to draw attention to it, which will appear deceptive to the reader.
TK: an illustration showing comparable meaning for a title -> connector sentence versus slightly changed meaning that is not acceptable.