How do characters (hero or non-hero) in the Greek epics The Iliad and The Odyssey respond to the notion of heroism?

The Iliad and The Odyssey, both written by the Ancient Greek poet Homer, are two of the most studied and most admired epic poems of Ancient Greek literature. The Iliad is one of the best historical accounts of the Trojan War, and it centers on the Ancient Greek hero Achilles. The Odyssey focuses around the Ancient Greek king of Ithaca, Odysseus, as he returns home after the fall of Troy. Odysseus plays an important role in The Iliad as well.
Almost every character in Homer’s epic poems, including the antagonists, deeply value the notions of honor and heroism, and I would even go as far as to say that they value honor more than life itself. Thus, heroes and non-heroes respond quite similarly to the notions of heroism and glory. Homer tells us that almost all of his characters were ready to give their lives for their country, for the greater good, or for a personal cause they believed in; for them, to be considered heroic was the greatest honor in the world. To die for something grand means that their sacrifice will be considered brave and noble and that they will always be regarded as heroes. This is definitely the case with Achilles in The Iliad, but perhaps not as much with Odysseus in The Odyssey, as the Greek king shamelessly values his own life above all else, even when his prophesied fate is to die.
The characters from both The Iliad and The Odyssey do not see death as punishment or defeat; instead, they see it as an act of heroism. Dying as a hero is honorable, and dying as a coward is considered shameful and disgraceful; in fact, to be remembered as a coward or a traitor and have a tarnished reputation is considered the worst possible failure by both Homer's protagonists and his antagonists.
The notions of heroism and honor are especially important to the heroes. Homeric heroes believe that heroism and glory can only be achieved by showing immense courage, strength, bravery, and dignity on the battlefield; they remain loyal to their countries, and they treat soldiers, victims, and enemies alike with respect. To be a hero for them means that they have to be responsible and disciplined and fulfill society’s expectations of them. Thus, they mold and shape their identities according to society’s perceptions and norms and play the role of the hero both at home and on the battlefield.
However, to be a hero also means they must not be arrogant and convinced that they are invincible; on the contrary, they have to be humble and honorable, especially before the mighty gods. Thus, arrogance, betrayal, and cowardice are the main qualities that separate the heroes from the villains.