%
\label{lem:yoneda-application}
There is a canonical bijection
\[
    \boxed{%
    \mathrm{Nat}\bigl(y(\tau^3),\, y(H_\tau)\bigr)
    \;\xleftrightarrow{\;\sim\;}
    \{\textup{$\omega$-germ transformers on } \tau^3
    \textup{ valued in } H_\tau\}.}
\]
Under this bijection,
a natural transformation
$\eta \colon y(\tau^3) \to y(H_\tau)$
corresponds to the $\omega$-germ transformer~$G_\eta$
defined by:
\begin{enumerate}
    \item[\textup{(i)}]
          At each stage~$k$,
          the component $G_{\eta,k} \colon
          \mathbb{Z}/P_k\mathbb{Z} \to H_\tau / P_k H_\tau$
          is defined by
          $G_{\eta,k}(a) := \pi_k\bigl(\eta_{\mathbb{Z}/P_k\mathbb{Z}}(\iota_k^a)\bigr)$,
          where $\iota_k^a \colon \mathbb{Z}/P_k\mathbb{Z} \to \tau^3$
          is the stage-$k$ probe selecting class~$a$.

    \item[\textup{(ii)}]
          The family $(G_{\eta,k})_{k \geq 1}$
          is tower-coherent:
          the naturality square
          applied to the projection
          $\mathbb{Z}/P_{k+1}\mathbb{Z}
          \to \mathbb{Z}/P_k\mathbb{Z}$
          ensures
          $\pi_{k+1 \to k}(G_{\eta,k+1}(b))
          = G_{\eta,k}(\pi_{k+1 \to k}(b))$
          for all $b \in \mathbb{Z}/P_{k+1}\mathbb{Z}$.

    \item[\textup{(iii)}]
          The tower-coherent family
          $(G_{\eta,k})_{k \geq 1}$
          defines an $\omega$-germ transformer
          in the sense of
          Book~I (I.D47--I.D49):
          a coherent system of stagewise maps
          whose inverse limit is a map
          $\tau^3 \to H_\tau$ at the $\omega$-level.
\end{enumerate}
Conversely, every $\omega$-germ transformer
$G = (G_k)_{k \geq 1}$
defines a natural transformation
$\eta_G$ by reversing the construction:
$(\eta_G)_P(\varphi) := G \circ \varphi$,
where the composition is interpreted stagewise.

The proof has two directions.

\smallskip
\noindent\textbf{Forward: natural transformation $\to$ $\omega$-germ transformer.}
Let $\eta \colon y(\tau^3) \to y(H_\tau)$
be a natural transformation.
By Proposition~\ref{prop:ch50-primorial-probes},
every probe factors through a finite stage,
so $\eta$ is determined by its action
on stage-$k$ probes for all~$k$.
At stage~$k$, the stage-$k$ probes
are indexed by $a \in \mathbb{Z}/P_k\mathbb{Z}$,
so $\eta$ at stage~$k$ defines a function
$G_{\eta,k} \colon \mathbb{Z}/P_k\mathbb{Z} \to H_\tau / P_k H_\tau$.
The naturality square for the projection
$\mathbb{Z}/P_{k+1}\mathbb{Z} \to \mathbb{Z}/P_k\mathbb{Z}$
gives tower coherence (item~(ii)).
The resulting tower-coherent family
is an $\omega$-germ transformer (item~(iii)).

\smallskip
\noindent\textbf{Backward: $\omega$-germ transformer $\to$ natural transformation.}
Let $G = (G_k)_{k \geq 1}$
be an $\omega$-germ transformer.
For any test object~$P$ and probe
$\varphi \colon P \to \tau^3$,
choose $k$ such that $\varphi$
factors through stage~$k$
(Proposition~\ref{prop:ch50-primorial-probes}):
$\varphi = \iota_k \circ \varphi_k$.
Define
\[
    (\eta_G)_P(\varphi) := \iota_k^{H_\tau} \circ G_k \circ \varphi_k,
\]
where $\iota_k^{H_\tau}$
is the stage-$k$ inclusion into~$H_\tau$.
This is well-defined (independent of the choice of~$k$)
by tower coherence of~$G$.
Naturality of~$\eta_G$
follows from the stagewise composition structure:
for $\psi \colon Q \to P$,
\begin{align*}
    (\eta_G)_Q(\varphi \circ \psi)
    &= G \circ (\varphi \circ \psi) \\
    &= (G \circ \varphi) \circ \psi \\
    &= (\eta_G)_P(\varphi) \circ \psi,
\end{align*}
where the middle step uses the associativity
of stagewise composition
(Theorem~\ref{thm:associativity}, II.T28,
Chapter~\ref{ch:composition-structure}).

\smallskip
\noindent\textbf{Bijectivity.}
The two maps are inverse to each other:
starting from~$\eta$, constructing~$G_\eta$,
then reconstructing $\eta_{G_\eta}$
recovers~$\eta$ on all probes
(because probes determine~$\eta$ completely
by the Yoneda embedding, II.T36).
Starting from~$G$, constructing~$\eta_G$,
then extracting $G_{\eta_G}$
recovers~$G$ at each stage
(by the probe decomposition,
Proposition~\ref{prop:ch50-primorial-probes}).