If Joe envied something, would it be proper to call that something "Joe's Envy"? Considering the usage in a team name but I am not sure if it makes sense.
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No. Envy is an emotion. To envy is to experience that emotion in relation to something. The object of envy is not envy.
However, from a literary perspective, you could still use "Joe's Envy" as a story or book title.
Just not as the object of that envy.