Essentially unique

In mathematics, the term essentially unique is used to indicate that while some object is not the only one that satisfies certain properties, all such objects are "the same" in some sense appropriate to the circumstances. This notion of "sameness" is often formalized using an equivalence relation.

A related notion is a universal property, where an object is not only essentially unique, but unique up to a unique isomorphism (meaning that it has trivial automorphism group). In general given two isomorphic examples of an essentially unique object, there is no natural (unique) isomorphism between them.

Most basically, there is an essentially unique set of any given cardinality, whether one labels the elements ${\displaystyle \{1,2,3\}}$ or ${\displaystyle \{a,b,c\}}$. In this case the non-uniqueness of the isomorphism (does one match 1 to a or to c?) is reflected in the symmetric group.

On the other hand, there is an essentially unique ordered set of any given finite cardinality: if one writes ${\displaystyle \{1<2<3\}}$ and ${\displaystyle \{a<b<c\}}$, then the only order-preserving isomorphism maps 1 to a, 2 to b, and 3 to c.

...
Wikipedia