In mathematics (especially in algebra or category theory), an object A having a certain property φ(A) is unique upto isomorphism if given any other object B such that φ(B), there exists an isomorphism f between A and B.

Effectively, that means that A is as good as it gets when it comes to φ-ness.

Examples of objects that are unique upto isomoprhisms abound in category theory. For instance, terminal objects are unique upto isomorphism (easy to prove), as are (by duality) initial objects. Indeed one often plays fast and loose with categories and implicitly identifies all objects upto ismoprhism: any two isomorphic objects are viewed through a sort of blurry lens in your mind and collapsed together.

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