Problem with functor Sym and Ord I am studying Category Theory on my own, and for now using Leinster’s book and class notes.  I have become stuck on a problem that seems important, but I simply cannot understand what he is saying when I read his answer.  My problem may be the language, and in one place his notation, but I don’t think my problem is really the math, per se.  I will much appreciate any help in understanding what is being said.
Here is the problem and answer from Leinster’s class notes.  The problem also appears on p. 39-40 of his book:


To me this answer makes no sense.  Here, Sym( ) is supposed to be a functor that maps the set X to the set of permutation of its elements.  But, the set of permutations on any set is the set of bijections from the set to itself.  Yet such bijections (morphisms in $ \textbf{B}$ ) are never considered.  To make sense we would need a map from the endo-bijective functions on X in $ \textbf{B}$ to the object Sym(X) in $ \textbf{Set}$ (which would not be a functor anyway since a functor must map objects to objects and morphisms to morphisms).  So what is going on here?  What exactly are the elements of Sym(X), and what is the nature of the functor to it from X? 
Also, I do not understand the notation relating to Sym(f) when he writes that $ \sigma $ maps to $ f\sigma f^{-1} $.  Is this a composition of three functions? Or What? (which again relates back to my question concerning the elements of Sym(X), since $ \sigma $ is apparently an element of Sym(X) .)
My problem with Ord( ) is similar, and I think that once I get Sym, then Ord will quickly follow.
To answer my question, perhaps a illustrative example is easiest.  If X={a,b} and Y={1,2}, what exactly is his answer saying explicitly for these two simple sets?
 A: I think the important thing is to keep careful track of the category that each construct belongs to.
$\newcommand{\Set}{\mathsf{Set}}$
$$
\begin{array}{r|ll}
\text{Category}&\text{Objects}&\text{Arrows}\\\hline
\mathcal{B} & \text{finite sets} & \text{bijections} \\
\Set & \text{sets} & \text{functions}
\end{array}
$$
In this problem, the set of permutations of $X$ is an object in the category $\Set$. Yes, every permutation is a bijection, but for this problem we're not considering it as a member of $\mathcal{B}$ in any way— all that matters is that a set of permutations is an object in some category called $\Set$.



*

*$\newcommand{\Set}{\mathsf{Set}} \newcommand{\func}[1]{\mathbf{#1}}$
$\mathcal{B}$ is the category of finite sets with bijections as maps.

*$\mathbf{Sym}:\mathcal{B}\rightarrow \Set$ operates on sets by sending a set $X$ to the set of permutations on $X$.

*To make $\func{Sym}$ into a functor, we have to define its behavior on arrows in a way that's compatible with its behavior on objects. The arrows in $\mathcal{B}$ are bijections between finite sets. We need to send each arrow $f:X\rightarrow Y$ in $\mathcal{B}$ to an arrow in $\Set$ between the permutations of $X$ and the permutations of $Y$ (these permutation-sets are considered as objects in $\Set$ !).
$$
\begin{array}{r|lll}
\text{Category}&\text{Obj}&\text{Arrows}&\text{Obj}\\\hline
\mathcal{B} & \text{finite set } X & \text{bijection }f & \text{finite set }Y \\
\mathbf{Sym}\downarrow && \\
\Set & \text{perms of }X & \text{fn between perm sets} & \text{perms of }Y\\
%% & =\mathbf{Sym}(X) & =\mathbf{Sym}(f) & =\mathbf{Sym}(Y)
\end{array}
$$
If $X$ and $Y$ are related by a bijection $f$, then there's a straightforward way to turn any permutation $\sigma$ on $X$ into a permutation on $Y$, namely: given an element $y$ to permute, convert it into an element $x = f^{-1}(y)$, then apply the $X$-permutation $x^\prime = \sigma(x)$, then convert it back $y^\prime = f(x^\prime)$. You can verify that this rule results in a well-defined permutation on $Y$.
Overall, if $X$ and $Y$ are sets related by a bijection $f$, we define $\mathbf{Sym}(f)$ to be a function which sends the set of permutations $\mathbf{Sym}(X)$ to the set of permutations $\mathbf{Sym}(Y)$ by mapping each individual permutation as $\sigma \mapsto f\circ \sigma \circ f^{-1}$.

One of the main takeaway points of the exercise is that, for any finite set $X$, the set of permutations of $X$ and the set of total orders of $X$ have the same number of elements. That is, they're bijective.
But despite this numerical similarity, they're not really structurally related: there's not an obvious non-arbitrary way to match up each permutation of $X$ with a total order of $X$. There are lots of ways to do so, of course— the sets are bijective. But none of them are natural.
This is formalized by considering the relationship between each set and its permutations / total orders as a functor, and showing that there's no natural transformation between permutations and total orders as functors.  That's the category-theoretic perspective on these constructs.
