Dajh Katzroy is the young son of Sazh Katzroy in the Final Fantasy XIII series. Prior to the events of Final Fantasy XIII, he was with his father at the Euride Gorge energy plant, a common tourist attraction for its scenery and field trip location for school children looking forward to learning about the role of the fal'Cie in society. The day they visited, however, the fal'Cie Kujata made Dajh a l'Cie to find the Pulse l'Cie that infiltrated the facility.
After visiting a fireworks display, Dajh was taken away by the Sanctum so that they might find out his focus that Kujata gave him. Many of his father's motivations throughout his journey are linked to the possibility of saving Dajh from becoming a Cie'th. Sazh cares for his son deeply, and prior to Dajh's incarceration by the Sanctum, had adopted a chocobo chick he intended to give to him as a present.
As it turns out, Dajh's focus was to locate Sazh and Vanille, as upon reuniting with his father, he immediately becomes crystal. The reunion was a trap staged by PSICOM officer Jihl Nabaat, who takes Sazh and Vanille into custody.
At the end of Final Fantasy XIII, Dajh is freed from his crystal stasis along with Serah and is able to reunite with his father.
Final Fantasy XIII-2
In Final Fantasy XIII-2, Dajh is aboard his father's airship when they accidentally fly into a Historia Crux and is separated from both his father and their chocobo chick. Sazh, meanwhile, ends up in the casino purgatory of Serendipity, and must win the opportunity to reunite with his son. After their reunion, they travel to the year 500 AF and participate in a battle against Caius Ballad.
Lightning Returns: Final Fantasy XIII
In Lightning Returns, after the Chaos is unleashed from Valhalla and floods Gran Pulse, turning it into the world of Nova Chrysalia, Dajh's soul is separated from his body and shatters into five fragments that become scattered around the world. Sazh spends the next five centuries searching for the fragments without luck, though he is given a special box by Lumina capable of reassembling the fragments and returning Dajh's soul to his body.
Lightning comes to their aid by searching the world and recovering all five of the fragments by following hints provided by Lumina and some assistance from Chocolina. When she returns them to Sazh, he attempts to return Dajh's soul, but the long centuries have made him grim and angry, preventing the process from working. Only by reconnecting with his old, joyful demeanor is Sazh able to return Dajh's soul and revive his son.
At the game's end, Dajh arrives with his father and the rest of humanity's souls to aid Lightning in the final battle against Bhunivelze. After the god is struck down, he travels to the new world with the rest of humanity, where he is able to be reborn.