Let $\mathcal{C}$ be a category.
- Existence. Assuming the axiom of choice, $\mathsf{Sk}\webleft (\mathcal{C}\webright )$ always exists.
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Pseudofunctoriality. The assignment $\mathcal{C}\mapsto \mathsf{Sk}\webleft (\mathcal{C}\webright )$ defines a pseudofunctor
\[ \mathsf{Sk}\colon \mathsf{Cats}_{\mathsf{2}}\to \mathsf{Cats}_{\mathsf{2}}. \]
- Uniqueness Up to Equivalence. Any two skeletons of $\mathcal{C}$ are equivalent.
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Inclusions of Skeletons Are Equivalences. The inclusion
\[ \iota _{\mathcal{C}}\colon \mathsf{Sk}\webleft (\mathcal{C}\webright )\hookrightarrow \mathcal{C} \]
of a skeleton of $\mathcal{C}$ into $\mathcal{C}$ is an equivalence of categories.
