The Trading button.

Trading was introduced in the 8/16/18 update. It allows you to trade any of your pets with others, or give them away for free. You must have at least 2 pets to be allowed to trade, due to being able to trade away your only pet for free.

Select the Trading button on the side menu. There, a list of players with trading enabled will show. You can request to trade pets with members.

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