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.
Famous quotes containing the words essentially and/or unique:
“The passion of love is essentially selfish, while motherhood widens the circle of our feelings.”
—Honoré De Balzac (1799–1850)
“An actor must communicate his author’s given message—comedy, tragedy, serio- comedy; then comes his unique moment, as he is confronted by the looked-for, yet at times unexpected, reaction of the audience. This split second is his; he is in command of his medium; the effect vanishes into thin air; but that moment has a power all its own and, like power in any form, is stimulating and alluring.”
—Eleanor Robson Belmont (1878–1979)