Often in mathematics, one encounters certain structures in which there are natural choices to be made. For example, there could be a 'god given' map relating two mathematical objects, in the sense that there are no arbitrary choices to be made in selecting this map.
The best example that comes to mind is the isomorphism between a vector space $V$ and its double dual $(V^{*})^{*}$: Given a finite dimensional vector space $V$, we can consider its dual vector space $V^{*}$. Since $\dim(V) = \dim(V^{*})$, we know that $V$ and $V^{*}$ are isomorphic. However, there is no distinguished (or natural) choice of isomorphism, one must choose a basis for $V$ to construct this isomorphism. Because of this fact, we cannot identify $V$ and $V^{*}$. On the other hand, there is a distinguished ('god given') choice of isomorphism between $V$ and $(V^{*})^{*}$. It is the isomorphism $\phi: V \to (V^{*})^{*}$, where given any $\alpha \in V^{*}$, and $v \in V$, $\phi(v)(\alpha) = \alpha(v)$. Because of this fact, we can identify $V$ and $(V^{*})^{*}$.
I read somewhere that natural transformations make precise this somewhat vague notion of natural/god-given maps between objects. In the example of vector spaces above, I know there is a natural isomorphism between the identity and $\phi$. So in this case, the concept of natural transformations makes the above natural identification of $V$ and $(V^{*})^{*}$ a bit more precise.
My question is whether this sort of thing happens in general: in the case where there is a natural choice to be made (for example, in choosing an isomorphism), can I expect to find a natural transformation? And to what extent does this help make precise the notion of non-arbitrariness/god-given.