Secure the Agency!
1st and 4th battles Rhyos and Vertigo
2nd and 5th battles Barbuta x3
3rd and 6th battles Flan Blanco and Barbuta

Winning all six battles in a row gets you Mission Complete! You receive the Pride of the Sword grid as a reward. Afterward in the Agency, an Al Bhed will give you the Berserker dressphere and an Al Bhed Primer.

Walkthrough:
Required: No
Description: Rout the fiends that have attacked the Travel Agency! There's no time to heal between battles, so you'll just have to withstand them all!
Objective: Win six consecutive battles