There are several moments in the text where it is clear that Odysseus wants to return home to Penelope and the life he led in Ithaca. Homecoming is something that causes pangs in him. This is seen in Book V, when Homer articulates that while Odysseus might possess everything, he lacks the one thing that truly matters to him:
By nights he would lie beside her, of necessity, in the hollow caverns, against his will, by one who was willing, but all the days he would sit upon the rocks, at the seaside, breaking his heart in tears and lamentation and sorrow as weeping tears he looked out over the barren water.
Odysseus might have everything he could want in terms of being with Calypso, but it is "barren" in comparison to what he yearns for at home. The "lamentation and sorrow" evident reflects how he just wants to return home to Penelope and the life he once lived as a mortal.
When Odysseus describes his own trial in Book IX, it is clear that all he really yearns for is what all mortals seek in terms of coming back home. For Odysseus, there is "nothing dearer" than this hope: "... but they could neither of them persuade me for there is nothing dearer to a man than his own country and his parents, and however splendid a home he may have in a foreign country, if it be far from father or mother, he does not care about it." Odysseus makes the argument that being a foreigner, a stranger in a strange land, is a haunting condition that strikes at the individual. It is one that compels the individual to only seek the comforts of home. This is echoed in Book V, when Odysseus speaks to how he is committed to simply wanting to return home to his wife a world where he is no longer a "stranger:"
...what I want and all my days I pine for is to go back to my house and see my day of homecoming. And if some god batters me far out on the wine-blue water, I will endure it, keeping a stubborn spirit inside me, for already I have suffered much and done much hard work on the waves and in the fighting. So let this adventure follow.
The spirit of endurance and struggle that exists as a part of his bravery is that eventually, Odysseus will return home. Being able to return home is what drives him and animates his very being in the world.