Skeleton of a category

Summary

A skeleton of a category $\mathcal{C}$ is a full subcategory that contains exactly one object from each isomorphism class of objects of $\mathcal{C}$. Under the axiom of choice every category has a skeleton, and the inclusion of the skeleton is an equivalence of categories. So a skeleton is a smallest model of the category up to equivalence, and two categories are equivalent exactly when their skeletons are isomorphic.