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categories: Skeleton of a category

Would you let me know when the category has an equivalent skeleton? (The 
definition of the skeleton subcategory that I have in mind is from
MacLane p91: a full subcategory such that for any object in the original 
category, there exists a unique isomorphic object in the
skeleton subcategory.) My question is mainly about when I can use the
choice axiom without causing contradiction. For instance, I heard that the
category of abelian groups doesn't have an equivalent skeleton

Thank you very much,