TITLE: Natural isomorphisms and the axiom of choice
QUESTION [16 upvotes]: The definitions of "natural transformation", "natural isomorphism between functors", and "natural isomorphism between objects" captures - among other things - the intuitive notion of "an isomorphism that does not depend on an arbitrary choice" (from Wikipedia). The standard example is a finite dimensional vector space $V$ being naturally isomorphic to its double dual $V$** because the isomorphism doesn't depend on the choice of basis.
What I wonder:

Does this informal notion of choice have to do
with the formal notion of choice in the axiom of
choice?
Is "being naturally
isomorphic" somehow related
to "being provably isomorphic from $ZF$ without $AC$"?
Or are these completely unrelated concepts?

REPLY [7 votes]: Short answer: The formal definition of “natural isomorphism” is completely unrelated to any notion of choosing, let alone the axiom of choice.
Silly answer: There are natural isomorphisms which depend on arbitrary choices. For example, under the axiom of global (!) choice, the category of sets is equivalent to the category of cardinals, which is a full subcategory of the category of sets. As such, every set is “naturally” isomorphic to its cardinal! Indeed, for each set $X$, fix a bijection $\eta_X : X \to \# X$, where $\# X$ is the cardinal of $X$. (If $X = \# X$, for simplicity we require $\eta_X = \text{id}_X$. This data suffices to specify a functor $\# : \textbf{Set} \to \textbf{Card}$, which acts on arrows $f : X \to Y$ by $\# f = \eta_Y \circ f \circ {\eta_X}^{-1}$. Let $U : \textbf{Card} \hookrightarrow \textbf{Set}$ be the inclusion. Then, $\#$ is left adjoint to $U$ with counit $\eta$, since by construction
$$\begin{matrix}
X & \xrightarrow{\eta_X} & \# X \newline
{\scriptstyle f} \big \downarrow & & \big\downarrow {\scriptstyle \# f} \newline
Y & \xrightarrow{\eta_Y} & \# Y
\end{matrix}$$
commutes for every arrow $f : X \to Y$, and obviously both functors are full and faithful. This shows that $\eta$ is the “natural” isomorphism we seek.