The "Lost Temple Cat" quest in Fire Emblem: Fortune's Weave is a kindness quest for Cai that offers significant renown points and gold upon completion. This quest becomes available during Chapter 6.
To initiate the "Lost Temple Cat" quest, speak with the Priest situated in front of Smyrno's temple in the Upper Temple Row. The quest is active between April 20th and May 20th and rewards players with 2,500 gold and 130 renown points.
Following the conversation with the Priest, you will be tasked with locating the lost cat. The only clue provided is that the cat "went down Temple Row and is now wandering the area near the arena."
Your first destination should be the Arena Plaza. Once you are positioned in front of the Arena Reception NPC, turn southeast. You will find a large fountain in the vicinity. Navigate around this fountain, and the cat will be situated behind it. A quest marker will appear above the cat as you approach.
After interacting with the cat, return to the Priest in the Upper Temple Row to report your findings and complete the quest, thereby receiving your rewards.