Hope! This is the kind of power you've got inside. And it's telling you not to give up. Trust me.
Days after the party arrives on Gran Pulse, they remain in Vallis Media with Hope opting to gather food with the Chocobo Chick. When the Chocobo Chick returns to camp in alarm, the party hurries to find Hope unconscious. That night, the party discusses the option of heading to Oerba to find a way to remove their brands.
Hope tells them to leave him behind as he is scared they will get hurt because of him and inadvertently summons Alexander. Lightning and Fang come to Hope's aid with Lightning telling Hope the Eidolon is not an ordeal but a gift, not there to kill his summoner, but to show him he has the power to get back on track and continue on.
Alexander initially targets Hope with physical attacks. Like all Eidolons, Alexander inflicts Doom on Hope at the beginning of the battle, and instead of being defeated, Hope must fully fill Alexander's Gestalt Bar. Libra will reveal hitting him to get chain bonuses will cause him to yield.
The player will need just three paradigms - Protection (SYN/MED/SEN), Mystic Tower (RAV/RAV/SEN), and Combat Clinic (MED/MED/SEN) - so one should be sure to create the paradigms before the start of the fight. The party should start off the fight with Protection Paradigm and cast Protect on everyone and follow up with Mystic Tower. When the party's health gets low they switch to Combat Clinic, and then back to Mystic Tower.
"Alexander" is a common western name of Greek origin, the original name being Ἀλέξανδρος (Alexandros), "Protector of Mankind". It may also be a reference to the famous Macedonian king, Alexander the Great.