$\newcommand\lZ {{\leftadj \Z}}$
$\newcommand\gG {{\rightcat G}}$
$\newcommand\lm {{\leftcat m}}$
$\newcommand\rn {{\rightcat n}}$
It could equally well have been titled "Orbits and the Yoneda lemma".
The connection between orbits and natural transformations is that,
given a monoid,
<I>the exponential law relating (a power to a product of natural numbers) to (an iterated power)
is precisely (an instance of naturality)</I>.
We assume we are working in the familiar category of abelian groups and group homomorphisms.
Let $G$ be an abelian group.
The group's binary operation allows us to define an action of (the integers $\lZ$) on (that group $G$) by exponentiation.
A careful development of this is in many texts on algebra;
e.g., <i>Algebra</i>, Third Edition, by Mac Lane and Birkhoff,
in its Chapter II, Groups, Section 3, Cyclic Groups (M-B II.3),
where they define exponentiation ("powers") in an arbitrary group $G$,
using the group's operations and recursion:
$$ g^0 = e, \kern3em g^{(n+1)} = g^n g, \kern3em g^{(-n)} = (g^n)^{(-1)} $$
Once that is done one may prove "exponential laws" giving relationships between
the exponentiation operation giving an action of $\lZ$ on $\gG$ and
the algebraic operations on $\lZ$ and $\gG$:
\[ \boxed{ \begin{array} {ccc|l|c} &&&& \text{M-B II.3} \\ \hline g^{(n+n')} & \leftadj{ \xlongequal [\text{sum}] {\text{integer}} } & g^n g^{n'} & \text{exponentiation by } \leftadj{ \text{(a sum of integers)} } & (6) \\ g^0 & \xlongequal{} & e \\ \hline \\ g^{(\source n \target p)} & \rightadj{ \xlongequal [\text{product}] {\text{integer}} } & (g^{\source n})^{\target p} & \text{exponentiation by } \rightadj{ \text{(a product of integers)} } & (7) \\ g^1 & \xlongequal{} & g \\ \hline \\ (\source g \target h)^n & \target{ \xlongequal [\text{operation}] {\text{group}} } & \source{g^n} \target{h^n} & { \textstyle \text{exponentiation of } \target{ \text{(a product of group elements)}} \atop \textstyle \text{(this is the only law that assumes } \target{ \text{$G$ is abelian} ) } } & (12) \\ e^n & \xlongequal{} & e \\ \end{array} } \]
Since $\Set$ is cartesian closed, the action may be given in three equivalent ways:
\[ \boxed{ \begin{array} {ccccccc|l} \lZ & \longrightarrow & [G,G] & : & n & \mapsto & {()}^n & \text{exponentiation by $n$} \\ G \times \lZ & \longrightarrow & G & : & \langle g,n \rangle & \mapsto & g^n & \text{the usual action map} \\ G & \longrightarrow & [\lZ,G] & : & g & \mapsto & \hat g = g^? & \text{the orbit of $g$} \\ \end{array} } \]
Here is a preliminary, incomplete diagram in that category:
\[ \boxed { \begin{array} {ccccccccccc|cc} \kern10em & 1 \\ && \llap{ \text{either $1$ or, more generally, $m$} } \searrow \\ &&& \llap{ \leftcat{ 1 \in {} } } \rightcat{ \hom {r_0} {\rightadj{\Z^×}} {r_0} } \rlap{ \red{ \xrightarrow [\kern11em] { {g^?} = {\alpha_{\rightcat{r_0}}} } } } &&&& \hom {} {\leftcat G} {\rightcat{r_0}} \rlap{ {} \ni \red{ \boxed{ \source 1\alpha = \leftcat g } } } &&&& {\source g} {\target h} \\ &&& \lower10ex{ \source{ \llap{\rightcat{ \hom {r_0} {\rightadj{\Z^×}} {n} } } \smash{\Bigg\downarrow} } } & \rightcat{ \searrow \rlap{ \kern-4em \hom {r_0} {\rightadj{\Z^×}} {{\leftcat n}{\rightcat p}} } } & \lower10ex{ {\red\alpha}_{\leftcat n} } && \lower10ex{ \smash{ \source{ \Bigg\downarrow \rlap{G_n = ()^n} } } } & \searrow \rlap{G_{(\leftcat n \rightcat p)} = ()^{(\leftcat n \rightcat p)} } \\ &&& \llap{ \leftcat{ 1+\cdots +1 = 1\cdot n = 1 \tensor n = n \in {} } } \rightcat{ \hom {r_0} {\rightadj{\Z^×}} {r} } \rlap{ \red{ \xrightarrow [\kern11em]{ {g^?} = {\alpha_{\rightcat{r}}} } } } &&&& \hom {} {\leftcat G} {\rightcat r} \rlap{ \ni \boxed{ \begin{array} {ccc|c} \red{ \big( \leftcat{ 1+\cdots +1 } \big) \alpha } & \xlongequal{\text{$n$-ary homomorphism}} & (1\red\alpha) \cdots (1\red\alpha) & \text{$n$-ary operations} \\ \red{ (\leftcat{1\cdot n}) \alpha } & \xlongequal [ \text{$\rightadj{\Z^×}$-natural transformation } 1{\red\alpha}_n ] {\text{$\leftadj\Z$-homomorphism}} && \text{$\rightadj{\Z^×}$-actions} \\ \red{ (\leftcat{1 \tensor n)} \alpha } \\ \red{ \boxed{\black n \alpha} } & \xlongequal{\text{definition of $\red\alpha$ (given $\leftcat g$)}} & \red{ \boxed{ \black{(1\red\alpha)^n} = {\leftcat g}^{n} } } \\ \hline {} \rlap{\kern-2em \text{the general homomorphism property:}} \\ \red{ (\black{n+n'}) \alpha } & \xlongequal{\text{homomorphism}} & (n \red\alpha) (n' \red\alpha) \\ g^{n+n'} & \xlongequal{\text{exponential law}} & g^n g^{n'} \\ \end{array} } } &&& \kern30em & ({\source g} {\target h})^n = {\source{g^n}} {\target{h^n}} \\ &&&& \rightcat{ \llap{ \hom {r_0} {\rightadj{\Z^×}} {p} } \searrow } && {\red\alpha}_{\rightcat p} && \target{ \searrow \rlap{G_p = ()^p} } \\ &&&&& \rightcat{ \hom {r_0} {\rightadj{\Z^×}} {r'} } \rlap{ \red{ \xrightarrow [\kern10em] { {{\leftcat g}^?} = {\alpha_{\rightcat{r'}}} } } } &&&& \hom {} {\leftcat G} { \rightcat{r'} } \\ \end{array} } \]
\[ \boxed { \begin{array} {} & & \red{ \big( \leftadj m \rightcat{ (\leftcat n p) } \big) \alpha_{\rightcat{r'}} } & \red{ \xlongequal { \leftadj m \alpha_{ (\leftcat n \rightcat p) } } } & ( \leftadj m {\red\alpha}_{\rightcat{r_0}} )^{ \rightcat { ( \leftcat n p ) } } & \kern1em \\ & & \llap{ \red{( \rightadj{\text{assoc.}} ) \alpha_{\rightcat{r'}} } } \Vert \\ \red{ \big( \leftcat n \rightcat p \big) \alpha_{\rightcat{r'}} } & \xlongequal{ \leftadj{m=1} } & \red{ \big( \leftcat{ (\leftadj m \leftcat n) } \rightcat p \big) \alpha_{\rightcat{r'}} } && \smash{ \raise1.5ex{\Bigg\Vert \rlap{ ( \leftadj m {\red\alpha}_{\rightcat{r_0}})^{ \rightadj{\text{assoc.}}} } } } \\ & \red{ (\leftadj m \leftcat n) \alpha_{\rightcat p} } \\ \rightcat{ \big( \red{ (\leftcat n) \alpha_{\rightcat{r}} } \big)^{\rightcat p} } & \xlongequal{ \leftadj{m=1} } & \rightcat{ \big( \red{ (\leftadj m \leftcat n) \alpha_{\rightcat{r}} } \big)^{\rightcat p} } & \red{ \xlongequal [ \rightcat{ ( \leftadj m {\red\alpha}_{\leftcat n} )^p } ] {} } & \rightcat{ \big( \leftcat{ (\leftadj m {\red\alpha}_{\rightcat{r_0}})^n } \big)^p } \\ \end{array} } \]
The next diagram is a work in progress.
It is being modified from a previous diagram.
\[ \boxed { \begin{array} {ccccccccccc|cc} \kern10em & 1 \\ && \llap{ \text{either $0$ or, more generally, $m$} } \searrow \\ &&& \llap{ \leftcat{ 0 \in {} } } \rightcat{ \hom {r_0} {\leftadj{\Z^+}} {r_0} } \rlap{ \red{ \xrightarrow [\kern11em] { {g^?} = {\alpha_{\rightcat{r_0}}} } } } &&&& \hom {} {\leftcat G} {\rightcat{r_0}} \rlap{ {} \ni \red{ \boxed{ \source 0 \alpha = \leftcat g } } } &&&& {\source g} {\target h} \\ &&& \lower10ex{ \source{ \llap{\rightcat{ \hom {r_0} {\leftadj{\Z^+}} {n} } } \smash{\Bigg\downarrow} } } & \rightcat{ \searrow \rlap{ \kern-5em \hom {r_0} {\leftadj{\Z^+}} {( \leftcat n + \rightcat p )} } } & \lower10ex{ {\red\alpha}_{\leftcat n} } && \lower10ex{ \smash{ \source{ \Bigg\downarrow \rlap{G_n = ()^n} } } } & \searrow \rlap{G_{(\leftcat n + \rightcat p)} = ()^{(\leftcat n + \rightcat p)} } \\ &&& \llap{ \leftcat{ 0 + n = n \in {} } } \rightcat{ \hom {r_0} {\leftadj{\Z^+}} {r} } \rlap{ \red{ \xrightarrow [\kern11em]{ {g^?} = {\alpha_{\rightcat{r}}} } } } &&&& \hom {} {\leftcat G} {\rightcat r} \rlap{ \ni \boxed{ \begin{array} {ccc|c} \red{ \big( \leftcat{ 1+\cdots +1 } \big) \alpha } & \xlongequal{\text{$n$-ary homomorphism}} & (1\red\alpha) \cdots (1\red\alpha) & \text{$n$-ary operations} \\ \red{ (\leftcat{1\cdot n}) \alpha } & \xlongequal [ \text{$\leftadj{\Z^+}$-natural transformation } 1{\red\alpha}_n ] {\text{$\lZ$-homomorphism}} && \text{$\leftadj{\Z^+}$-actions} \\ \red{ (\leftcat{1 \tensor n)} \alpha } \\ \red{ \boxed{\black n \alpha} } & \xlongequal{\text{definition of $\red\alpha$ (given $\leftcat g$)}} & \red{ \boxed{ \black{(1\red\alpha)^n} = {\leftcat g}^{n} } } \\ \hline {} \rlap{\kern-2em \text{the general homomorphism property:}} \\ \red{ (\black{n+n'}) \alpha } & \xlongequal{\text{homomorphism}} & (n \red\alpha) (n' \red\alpha) \\ g^{n+n'} & \xlongequal{\text{exponential law}} & g^n g^{n'} \\ \end{array} } } &&& \kern30em & ({\source g} {\target h})^n = {\source{g^n}} {\target{h^n}} \\ &&&& \rightcat{ \llap{ \hom {r_0} {\leftadj{\Z^+}} {p} } \searrow } && {\red\alpha}_{\rightcat p} && \target{ \searrow \rlap{G_p = ()^p} } \\ &&&&& \rightcat{ \hom {r_0} {\leftadj{\Z^+}} {r'} } \rlap{ \red{ \xrightarrow [\kern10em] { {{\leftcat g}^?} = {\alpha_{\rightcat{r'}}} } } } &&&& \hom {} {\leftcat G} { \rightcat{r'} } \\ \end{array} } \]
------
Proofs by induction of (the exponential laws for powers),
using (the definition of powers by recursion).
\[ \boxed{ \begin{array} {ccc|ccc} a^{(\lm + \rn)\sigma} & \xlongequal { \text{defn. $a^{(\lm+\rn)\sigma}$} } & a a^{(\lm + \rn)} & \boxed{ a^{\big(\lm(\rn\sigma)\big)} } & \xlongequal { \text{defn. $\lm(\rn\sigma)$} } & a^{(\lm\rn + m)} \\ \llap{ \text{defn. $\lm+(\rn\sigma)$} } \Vert && \Vert \rlap{ \text{ind.} } &&& \Vert \rlap{ \text{homo. for plus} } \\ \boxed{ a^{\lm+(\rn\sigma)} } && a a^\lm a^\rn &&& a^{\lm\rn} a^\lm \\ && \Vert \rlap{ \text{comm.} } &&& \Vert \rlap{ \text{ind.}} \\ \boxed{ a^\lm a^{\rn \sigma} } & \xlongequal [\text{defn. $a^{\rn\sigma}$}] {} & a^\lm a a^\rn & \boxed{ (a^\lm)^{(\rn\sigma)} } & \xlongequal [ \text{defn. $(a^\lm)^{(\rn\sigma)}$} ] {} & (a^\lm)^\rn a^\lm \\ \end{array} } \]
