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Lets take a category of sets and functions between them. So there are (infinitely?) many terminal objects. As far as I understand, it's no so hard to prove that any two terminal objects are isomorphic to each other.

Concept of being "unique up to isomorphism" still seems a bit unclear for me, so I'm wondering wether it is technically correct to say that those singletons-terminal objects are "unique up to isomorphism". It feels like I can go further and say that those are unique up to unique isomorphism, because the definition a of terminal object implies that no other isomorphism can take place simply becase no more arrows in-between any two terminal objects can appear.

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  • $\begingroup$ Yes, all objects defined by universal properties are unique up to unique morphism. $\endgroup$ May 3 '18 at 6:49
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    $\begingroup$ Some things are unique up to non-unique isomorphism also. Splitting fields for example $\endgroup$
    – Elliot G
    May 3 '18 at 6:54
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Your intuition is good. "Unique up to isomorphism," is usually weaker than what you want to show in most contexts in that there will also be a uniqueness statement for the isomorphisms themselves. In the case of terminal objects, you do literally get that there is a unique isomorphism between any two terminal objects, but in general you might not get something quite so strong.

Consider, for example, the cartesian product of sets $A \times B$. There are multiple sets satisfying this property, and they're all isomorphic, but there may be more than one isomorphism between them. In particular, any nontrivial automorphism of $A$ gives rise to an automorphism of $A \times B$ that is not the identity. The right uniqueness property for the isomorphisms, here, is that there is a unique isomorphism between any two realizations of $A \times B$ that commutes with the projections to $A$ and $B$.

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