Mulberry — Food safety facts
Safe to share · Cats · full breakdown on the linked food page
Ripe berries are safe in small numbers. Unripe fruit and the leaves are not.
Short answer
Yes, you only need to focus on the cat, and it's fine. Ripe mulberries are safe for cats in small amounts, so a few berries are not a concern. The main thing is to keep unripe fruit and leaves away from all pets, as those parts are not safe. Just offer your cat some fresh water and watch for any unusual stomach upset, though it's very unlikely with ripe fruit.