Equivalence of categories

Categories are equivalent iff they have isomorphic skeleta

Let be categories with skeleta respectively. Then assuming the Axiom of Choice, are equivalent categories iff their skeleta are isomorphic categories, cat i.e.

Proof

It suffices to show every category is equivalent to its skeleton, since the full result follows from Skeletal categories are equivalent iff they are isomorphic and transitivity of equivalence. Let be the inclusion functor. We construct a functor which maps objects to their unique isomorphic representative. For any invoke \gls{ac} to fix an isomorphism , and for a general define . Then

commutes whence is a natural isomorphism. Therefore and .

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