Unique up to unique isomorphism If an object $X$ has a non-trivial automorphism $g$, for any isomorphism $f$ with an object $Y$ there is another isomorphism $f \circ g$ between $X$ and $Y$, so there is not a unique isomorphism between $X$ and $Y$.

Does the second "unique" in "unique up
  to unique isomorphism" mean nothing
  else than that the object in question
  has no automorphism than the identity?

If the answer is positive: Why is this second - and somehow independent - fact so strongly interwoven (terminologically) with the fact of being unique up to isomorphism?
 A: When people say, for example, that the product $X \times Y$ of two objects $X, Y$ is unique up to unique isomorphism, that doesn't mean that $X \times Y$, as an object of the category, has no non-trivial automorphisms; it's easy to find examples where this is blatantly false. It means that if you have two objects $A, B$ which are both products of $X$ and $Y$ in the sense that they come with distinguished projection maps to $X$ and $Y$ satisfying the universal property, then there is a unique isomorphism $A \to B$ compatible with the projection maps. The projection maps are part of the data that defines a product, and in particular it is possible for the same object to be a product of $A$ and $B$ in two different ways (in the sense that the projection maps are different): those different ways are then related by an automorphism of the object.
Another way to say this is to say that a product is a terminal object in a certain category of cones, and as an object of this category, it follows that the product has no non-trivial automorphisms because, for any terminal object $1$, there is a unique map $1 \to 1$, which must be the identity.
