As part of their research, authors of The Leadership Challenge, James Kouzes and Barry Posner, surveyed over 15,000 people and asked them how they would like their leaders to behave. The overwhelming first-choice was "honesty". Honesty was followed by the ability to be "forward-looking", the ability to be "inspiring", and "competence".
Kouzes and Posner point out that the qualities of being honest, inspiring and competent are what communications researchers refer to as "source credibility". The root word of credibility is "credo" which means "I trust" or, "I believe". When followers look for the above four qualities in their leaders, they are doing a mental credit check on them.
They are looking for the leader's credibility.