Categories are equivalent iff they have isomorphic skeleta

Let $𝖢, 𝖣$ be categories with skeleta $S k 𝖢, S k 𝖣$ respectively. Then assuming the Axiom of Choice, $𝖢, 𝖣$ are equivalent categories iff their skeleta are isomorphic categories, #m/thm/cat/evil i.e.

𝖢 ≃ 𝖣 ⟺ S k 𝖢 ≅ S k 𝖣

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 $𝐼 : S k (𝖢) ↪ 𝖢$ be the inclusion functor. We construct a functor $𝐹 : 𝖢 ↠ S k (𝖢)$ which maps objects to their unique isomorphic representative. For any $𝑌 \in 𝖢$ invoke AC to fix an isomorphism $𝜑 𝑋 : 𝑋 \to 𝐹 𝑌$ , and for a general $𝑓 : 𝑋 \to 𝑌$ define $𝐹 𝑓 = 𝜑 𝑌 𝑓 𝜑 - 1 𝑋$ . Then

commutes whence $𝜑 : 1 \Rightarrow 𝐼 𝐹 : 𝖢 \to 𝖢$ is a natural isomorphism. Therefore $𝐼 𝐹 ≃ 1 𝖢$ and $𝐹 𝐼 = 1 𝖣$ .

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