Skeleton of a category

Remarks

A category is skeletal if isomorphic objects in it are equal; a skeleton is a skeletal full subcategory that meets every isomorphism class. Under the axiom of choice every category has a skeleton, and its inclusion is fully faithful and essentially surjective. So the inclusion is an equivalence of categories by the criterion for equivalences. Any two skeletons of $\mathcal{C}$ are isomorphic categories. Example: in the category of finite sets, the full subcategory on the sets $[n] = \set{1, \dots, n}$, $n \ge 0$, with $[0] = \emptyset$, is a skeleton. A finite set with $n$ elements is in bijection with $[n]$, and $[m]$ and $[n]$ are not in bijection when $m \ne n$. Non-example: the full subcategory on the sets $[n]$ together with $\set{a, b}$ is not a skeleton, since $\set{a, b}$ and $[2]$ are two of its objects in the same isomorphism class.