%
\label{def:pre-yoneda}
The \textbf{Pre-Yoneda embedding} is the map
\[
    \boxed{%
    y \;:\;
    \mathrm{Hol}_\tau(\tau^3,\, H_\tau)
    \;\hookrightarrow\;
    d(\tau^3),
    \qquad
    y(f) \;:=\; [G_f],}
\]
defined as follows.
Let $f \in \mathrm{Hol}_\tau(\tau^3, H_\tau)$
be a $\tau$-holomorphic function.
By the Mutual Determination Theorem (II.T27),
$f$ is uniquely determined by its $\omega$-germ
$[f] = [(f_k)_{k \geq 1}]$.
The $\omega$-germ transformer $G_f$
(II.D37, Chapter~\ref{ch:evolution-operator})
is the coherent family of stage maps
$\{(G_f)_k\}_{k \geq 1}$
defined by the action of $f$
on the finite stages:
$(G_f)_k \colon \mathbb{Z}/P_k\mathbb{Z} \to H_\tau$
is the stage-$k$ component of $f$.

The map $y$ sends the holomorphic function $f$
to its $\omega$-germ class $[G_f] \in d(\tau^3)$.