Significance of unique isomorphism In his answer to Unique up to unique isomorphism, Qiaochu Yuan explains quite well what is meant by something being "unique up to a unique isomorphism", but I'm a bit perplexed by the significance of the uniqueness of this isomorphism. What makes "unique up to unique isomorphism" more useful than merely "unique up to isomorphism"?
 A: "Unique up to unique isomorphism" is significant because not only is the object itself uniquely identified, but the individual elements are as well.  
For example, $\mathbb{Z}$ as an additive group is not unique up to unique isomorphism, because we cannot distinguish 1 from -1.  This means that any place a group isomorphic to $\mathbb{Z}$ arises, we will always have a choice of generator.  In the absence of additional information, there will be no natural way to decide which element is 1 and which is -1.
However, $\mathbb{Z}$ as a ring is unique up to unique isomorphism.  With multiplication, we can distinguish 1 from -1.  Whenever a ring isomorphic to $\mathbb{Z}$ arises, not only do we identify the ring itself, but also individual elements we can label 0,1,2,3,... and -1,-2,-3,...
Edit: Here is perhaps a better example.  All vector spaces over a fixed field $F$ of a fixed dimension $n$ are isomorphic.  However, this isomorphism is highly non-unique, relying on a choice of basis.  This tells us that we should not usually think of any such vector space $V$ as simply being elements of $F^n$, because there is no natural choice of which object of $V$ should be (1,0,...,0), etc.
