Hopefully, these warnings would include a link to the trade that would be canceled, and you could turn off trade warnings for certain groups so they wouldn't be horribly annoying for the users who run adoption centers.
Now, a number of people have pointed out that when you send a trade, it tells you if a pet is offered in another trade. This is true, but not what this suggestion is about. This suggestion is about when you accept a trade someone else sent you. Because other users can't tell what pet's you've offered in trades, they will often offer on those pets, and the only way you can tell if they're in another trade is when the other trade is autocanceled.
Edit: People have suggested that instead of having a pop-up, there would be a little notice, like this:
(Credit to Shenzai for the image)
(Another edit: I am aware the poll options are a little strange. I made this suggestion six years ago, and I don't want to risk misrepresenting what people voted for by changing the options.)
