%
\label{thm:omega-germs-holomorphic}
There is a canonical bijection
\[
    \boxed{%
    \{\textup{$\omega$-germ transformers on } \tau^3\}
    \;\xleftrightarrow[\Psi]{\;\sim\;}
    \mathrm{Hol}_\tau(\tau^3, H_\tau).}
\]
Explicitly:
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{Forward} ($\omega$-germ $\to$ holomorphic).
          Given an $\omega$-germ transformer
          $G = (G_k)_{k \geq 1}$,
          the Yoneda Application
          \textup{(Lemma~\ref{lem:yoneda-application}, II.L14)}
          produces a natural transformation
          $\eta_G \colon y(\tau^3) \to y(H_\tau)$.
          The Yoneda embedding
          \textup{(Theorem~\ref{thm:yoneda-embedding}, II.T36)}
          then produces a unique morphism
          $f_G \in \Hom_\tau(\tau^3, H_\tau)$
          with $y(f_G) = \eta_G$.
          This $f_G$ is $\tau$-holomorphic
          by the characterization theorem
          \textup{(II.T33)}.

    \item[\textup{(ii)}]
          \textbf{Backward} (holomorphic $\to$ $\omega$-germ).
          Given $f \in \mathrm{Hol}_\tau(\tau^3, H_\tau)$,
          the Pre-Yoneda embedding
          \textup{(Definition~\ref{def:pre-yoneda}, II.D50,
          Chapter~\ref{ch:pre-yoneda})}
          produces $y(f) \colon y(\tau^3) \to y(H_\tau)$.
          The Yoneda Application
          \textup{(Lemma~\ref{lem:yoneda-application})}
          then extracts the $\omega$-germ transformer
          $G_f = (G_{f,k})_{k \geq 1}$
          associated to~$y(f)$.

    \item[\textup{(iii)}]
          \textbf{Bijectivity.}
          The maps $G \mapsto f_G$ and $f \mapsto G_f$
          are inverse to each other.
\end{enumerate}

The proof assembles three previously established bijections.

\smallskip
\noindent\textbf{Step 1: Yoneda embedding.}
By the Yoneda embedding theorem (II.T36),
the map
\[
    y \colon \Hom_\tau(\tau^3, H_\tau)
    \;\xrightarrow{\;\sim\;}
    \mathrm{Nat}\bigl(y(\tau^3),\, y(H_\tau)\bigr)
\]
is a bijection.
Full faithfulness of~$y$
gives both injectivity and surjectivity.
This is the abstract Yoneda bijection,
valid for any pair of objects in~$\tau$.

\smallskip
\noindent\textbf{Step 2: Yoneda Application.}
By Lemma~\ref{lem:yoneda-application} (II.L14),
the map
\[
    \mathrm{Nat}\bigl(y(\tau^3),\, y(H_\tau)\bigr)
    \;\xrightarrow{\;\sim\;}
    \{\text{$\omega$-germ transformers on } \tau^3\}
\]
is a bijection.
This step uses the specific profinite structure
of~$\tau^3$ (primorial probe decomposition)
and the identification of probe naturality
with stagewise naturality of $\omega$-germs
(II.R12, II.L13).

\smallskip
\noindent\textbf{Step 3: Characterization.}
By the characterization theorem (II.T33,
Chapter~\ref{ch:three-lemma-chain}),
$\Hom_\tau(\tau^3, H_\tau)
= \mathrm{Hol}_\tau(\tau^3, H_\tau)$:
the morphisms from~$\tau^3$ to~$H_\tau$
in~$\tau$ are exactly the $\tau$-holomorphic functions.
This identification is given by the 3-lemma chain
(II.L08--II.L10):
every such morphism is idempotent-supported,
and every idempotent-supported function
is $\tau$-holomorphic.

\smallskip
\noindent\textbf{Composition.}
Composing Steps~1 and~2 gives a bijection
\[
    \Psi \colon
    \mathrm{Hol}_\tau(\tau^3, H_\tau)
    \;\xrightarrow[\text{Step 1}]{\;y\;}
    \mathrm{Nat}\bigl(y(\tau^3),\, y(H_\tau)\bigr)
    \;\xrightarrow[\text{Step 2}]{\;\sim\;}
    \{\text{$\omega$-germ transformers}\}.
\]
The inverse is the reverse composition
$\Psi^{-1} \colon G \mapsto y^{-1}(\eta_G) = f_G$.
Since each step is a bijection,
the composition is a bijection.

\smallskip
\noindent\textbf{Bijectivity} follows from the bijectivity
of each factor.