Free groups: unique up to unique isomorphism A free group $F_S$ over a set $S$ is unique up to unique isomorphism. After reading this answer here I think this means that if $F^\prime_S$ is another free group over $S$ then there exists exactly one isomorphism $\varphi : F_S \to F_S^\prime$ such that $\varphi \circ i = i^\prime$ where $i: S \to F_S$ and $i^\prime : S \to F^\prime_S$ are the inclusions.
Why is uniqueness of the isomorphism interesting at all? Wouldn't it be good enough to know that all free groups over $S$ are isomorphic to define free groups?
 A: One reason you are interested in "unique up to unique isomorphism" is that it makes things more "canonical."
Consider for example the following true fact: every $n$-dimensional vector space over $\mathbb{R}$ is isomorphic to $\mathbb{R}^n$.
However, except for $n=0$, the isomorphism is neither unique nor "canonical": if I go home and do computations based on my favorite isomorphism and I tell you the answer, and you go home and you do your computations based on your favorite isomorphism, and then we come back tomorrow and describe our results, in order to translate one result into "the language of the other" we need to find out what my favorite isomorphism is, what your favorite isomorphism is, and perform the entire translations in order to compare "my" translation into $\mathbb{R}^n$ with yours and find out if we are both correct.
However, when there is a canonical choice of isomorphism, then we can simply both go home and translate (via this unique isomorphism), and then present the results the next day. They must match, because there is essentially only one way to translate. 
(Canonicity is very useful; for instance, that is one reason why, still talking about linear algebra, the isomorphism in finite dimension between a vector space and its double dual is so much more important than the isomorphism between a vector space and its dual. The former is canonical, while the latter is not.)
From a point of view of category theory, uniqueness up to unique isomorphism allows for easy functorial properties to be derived, whereas merely isomorphic means a lot more book-keeping is needed, and it may be impossible to state a "coordinate-free" functorial property (where I'm thinking of 'coordinate-free' somewhat fuzzily to refer to not having to invoke specific, arbitrarily chosen isomorphisms to state or prove them). 
A: Uniqueness up to unique isomorphism implies that there is no automorphism $f:F_S\to F_S$ which is the identity on the image of $i:S\to F_S$.
