Since part of Cai’s gimmick involves side quests that have him running errands around Dagsion to be kind, there can be a number of tasks that don’t even force you to leave the capital city. Quite a few involve finding characters like a child who wandered away from her grandmother or three women with brooms. The Fire Emblem: Fortune’s Weave “Lost Temple Cat” quest involves finding a missing animal for one of the Smyrnos Temple priests, and here’s where you can find the missing kitty.
Picking up the “Lost Temple Cat” quest is easy in Fire Emblem: Fortune’s Weave, given the Priest Caring for Cats is right outside the entrance to the Smyrnos Temple, and that person even gives you a hint as to where the animal went. We know going in that it will be in the vicinity of the city’s Arena. The person also mentions it is a completely brown cat.
Here’s where you can find the Priest Caring for Cats.

As for finding the cat, it is near the Arena! However, that isn’t exactly where the “Lost Temple Cat” is in Fire Emblem: Fortune’s Weave. It’s actually behind the fountain that is closer to the elevators to the lower part of town. If you walk around the back of the water feature, you’ll see a brown cat sitting on the steps. Unlike other animals, this one will have a speech bubble above it and you can interact with it by pressing “A.”
Here’s where you can find the lost cat in Fire Emblem: Fortune’s Weave.

After Cai talks to the cat, it will head home. You can then return to the Priest Caring for Cats to report the results in exchange for 2,500 gold and 130 Renown.
Fire Emblem: Fortune’s Weave is available on the Switch 2.