%
\label{lem:probe-yoneda}
Let $\varphi \colon A \to B$ be a morphism in $\tau$.
The following conditions are equivalent:
\begin{enumerate}
    \item[\textup{(PN)}]
          $\varphi$ satisfies probe naturality
          (Definition~\ref{def:probe-naturality}).
    \item[\textup{(Hol)}]
          $\varphi$ is $\tau$-holomorphic
          in the sense of the enriched structure:
          the natural transformation $y(\varphi)$ respects
          the split-complex decomposition
          of each Hom object,
          i.e., $y(\varphi)$ preserves
          the bipolar sector structure (II.P11).
    \item[\textup{(Yon)}]
          The natural transformation $y(\varphi)$
          is an element of
          $\mathrm{Nat}(y(A), y(B))$
          that is determined by $\varphi$
          via the Yoneda correspondence:
          evaluation at $\id_A$ recovers $\varphi$.
\end{enumerate}

\textbf{(PN) $\Rightarrow$ (Hol).}\;
Assume $\varphi$ satisfies probe naturality.
By the commutativity of the diagram
in Definition~\ref{def:probe-naturality},
postcomposition by $\varphi$
intertwines with precomposition by any $h$.
In particular, taking $P$ to be the $\tau$-objects
on which the bipolar decomposition
$[P, A] = e_+ \cdot [P, A]_+ + e_- \cdot [P, A]_-$
lives:
the postcomposition map $\varphi_*$
must carry $e_+ \cdot [P, A]_+$ into $e_+ \cdot [P, B]_+$
and $e_- \cdot [P, A]_-$ into $e_- \cdot [P, B]_-$,
because precomposition by any probe $h$
preserves the bipolar splitting
(II.P11 applied to $h^*$),
and the commutativity of the naturality square
forces $\varphi_*$ to respect the same splitting.
Thus $y(\varphi)$ preserves the bipolar sector structure.

\medskip
\textbf{(Hol) $\Rightarrow$ (Yon).}\;
Assume $y(\varphi)$ preserves the bipolar sector structure.
We must show that the natural transformation $y(\varphi)$
is determined by its value at $\id_A$.
Evaluate $y(\varphi)$ at $P = A$ and $f = \id_A$:
\[
    y(\varphi)_A(\id_A) \;=\; \varphi \circ \id_A \;=\; \varphi.
\]
For any other probe $g \colon P \to A$:
\[
    y(\varphi)_P(g)
    \;=\; \varphi \circ g
    \;=\; y(\varphi)_A(\id_A) \circ g
    \;=\; g^*\bigl(y(\varphi)_A(\id_A)\bigr).
\]
The last equality holds because $y(\varphi)$
is a natural transformation:
evaluating at $\id_A$ and then precomposing with $g$
equals evaluating at $g$ directly.
This is the enriched Yoneda bijection:
$y(\varphi)$ is determined by $\varphi = y(\varphi)_A(\id_A)$.

The bipolar preservation hypothesis ensures
that this determination respects the enriched structure ---
the bijection $\mathrm{Nat}(y(A), y(B)) \cong [A, B]$
is an isomorphism of $\tau$-objects
(not merely a set bijection),
because both sides carry the bipolar decomposition
and the isomorphism preserves it.

\medskip
\textbf{(Yon) $\Rightarrow$ (PN).}\;
Assume the Yoneda correspondence holds:
$y(\varphi)$ is determined by $\varphi$ via evaluation at $\id_A$.
For any $h \colon P' \to P$ and $g \colon P \to A$:
\[
    h^*\bigl(y(\varphi)_P(g)\bigr)
    \;=\; h^*(\varphi \circ g)
    \;=\; (\varphi \circ g) \circ h
    \;=\; \varphi \circ (g \circ h).
\]
The last equality uses associativity in $\tau$
(II.T28, earned from the program monoid I.P02).
On the other hand:
\[
    y(\varphi)_{P'}(h^*(g))
    \;=\; y(\varphi)_{P'}(g \circ h)
    \;=\; \varphi \circ (g \circ h).
\]
Both sides agree,
so the naturality square commutes.