"There are three things which are stately in their march, even four which are stately when they walk
The lion which is mighty among beasts and does not retreat before any
The strutting rooster, the male goat, and the king when his army is with him"
This is the Hebrew word yatab which means to be good, well, glad, or pleasing. The idea clearly seems to be to bring together four evidences of pride and superiority in which an animal or person will not back down.
This is to teach a leader that you cannot get another leader to acknowledge that you were right and they were wrong, thereby backing down except in private. They will not accept this level of degradation in a public forum.
The key idea is what is said about the lion. He does not retreat before any. Too often we try and show we are right in a public way which stiffens our oppositions standing against any change. We need to approach another leader in private and lay out the case. Because when a leader has his troops, his assistants, and/or his followers with him, even if he is wrong, he will not admit it.
In one sense God is saying: If you want to be stately or powerful in your walk as a leader, you must realize how to try and change another person's mind. It is not to argue with them in public.
This is the Hebrew word melek which means king. This is the supreme leader in a country, but the lessons that Solomon and the other writers bring about is that this level of leadership is clearly meant to be used by all kinds of leaders.
Don't miss the lesson. You don't get someone else to see things your way or to give in to your ideas by challenging them in public or asking them to publicly to change, just as the next few verses spell out how you should not exalt yourself which is a great danger with a strong leader. So you should not ask a person to publicly back down. Find a way to be in agreement with them. Find a way that they win or can at least save face. Help the other leader see how what you want is really what they want but from a different angle.