The connector sentence is simply a way of making the wording flow better when you take an idea and use it to support an argument. Therefore, it must have the same meaning as the idea'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 idea was about. Or it might appear that a popular idea was misused to draw attention to an argument, which will appear deceptively to the reader.
TK: an illustration showing comparable meaning for a title -> connector sentence versus slightly changed meaning that is not acceptable.