In Categories for the Working Mathematician 2nd edition, page 17, Mac Lane indicates (I could verify it) that the isomorphism between a finite abelian group $G$ and its characters group $DG = Hom(G, \mathbb{T})$ (where $\mathbb{T}$ is the group of complex numbers of modulus 1) is not a natural transformation in $\mathbf{Ab}_f$ (category of finite abelian groups).
But he goes on and modifies the definition of arrows in $DG$ so that now in $D'G$ the arrows are $D'f = Df^{-1}$, and correspondingly restricts the arrows to only group isomorphisms. So the new category is $\mathbf{Ab}_{f,i}$, with objects the finite abelian groups and with arrows the group isomorphisms.
So, on $\mathbf{Ab}_{f,i}$, the modified $I \rightarrow D'$ is now a natural transformation, as one can check by its effect on identity and on composition of arrows (may have I overlooked a problem, actually?).
But Mac Lane nonetheless states that "it is not natural in the sense of our definition", and by that I understand that he makes reference to "the definition of $\tau$ depends on no artificial choice of bases, generators, or the like" earlier in the page, and to "this isomorphism depends on a representation of $G$ as a direct product of cyclic groups and so cannot be natural" later.
I cannot understand in precise terms why this is not a natural transformation. Especially since it looks like one based on the formal definition of a natural transformation. Is the author actually thinking of something stronger than a natural transformation, like a n.t. with a universal property, i.e. a universal object for some diagram?
Edit:
In reply to a comment, here is how the natural transformation is defined.
Let $G$ be a finite abelian group. There are $|G|$ possible isomorphisms $\phi:G \rightarrow D'G$, so just choose one. Call it $\phi_G$.
Then, for any other group $F$ isomorphic to $G$ (so that there exists an arrow $f:G\rightarrow F$), use the functoriality of $D'$ to define the natural transformation applied to $F$. So, $\phi_F=D'f\circ\phi_G$. This defines completely $\phi$ on all groups isomorphic to $G$.
So, to define $\phi$ as a natural transformation, one has to make a choice of one isomorphism for each class of isomorphic groups in $\mathbf{Ab}_{f,i}$.
This need of a choice, however, does not prevent the result from being a natural transformation. For example, in $\mathbf{Set}$ the cartesian product, equal to the categorical product, of a fixed object $A$ with an other, $X \rightarrow A\times X$ is natural in $X$, so there is a natural transformation ($\cdot \rightarrow A\times \cdot$), despite the fact that cartesian product is defined only up to isomorphism. Indeed, one can define for example $(a, b) := \{a, \{a, b\}\}$, or $(a, b) := \{\{a, b\}, b\}$. The choice is free but it has to be the same for all objects in order to preserve the functoriality of ($A\times\cdot$). In this case, the category is connected, because there is at least one arrow between each two objects, so the requirement of functoriality let us do only one choice for all. In the case of $\mathbf{Ab}_{f,i}$ we needed to do one choice for each connected component.