%
\label{thm:presheaf-characterization}
\[
    \boxed{%
    \mathrm{HolFun}
    \;=\;
    \mathrm{Nat}(\mathcal{F},\, \mathcal{F}_{\jj})
    \;\cap\;
    \mathrm{D\text{-}Hol}}
\]
A $\tau$-holomorphic function
(Definition~\ref{def:holfun}, I.D47)
is precisely a natural transformation
from the source presheaf $\mathcal{F}$
to the split-complex presheaf $\mathcal{F}_{\jj}$
(Definition~\ref{def:primorial-presheaf}, I.D83)
that additionally satisfies D-holomorphy
(sector independence,
Proposition~\ref{prop:sector-independence}, I.P22).

$(\subseteq)$\;
Let $T \in \mathrm{HolFun}$.
Tower coherence
(Definition~\ref{def:tower-coherence}, I.D46)
states that for all $k \leq \ell$:
\[
    \pi_{\ell \to k}\bigl(T_\ell(t)\bigr)
    \;=\;
    T_k\bigl(\pi_{\ell \to k}(t)\bigr).
\]
This is the naturality square for $T : \mathcal{F} \Rightarrow \mathcal{F}_{\jj}$:
\[
    \begin{array}{ccc}
        \mathbb{Z}/M_\ell\mathbb{Z}
            & \xrightarrow{T_\ell}
            & \mathbb{Z}/M_\ell\mathbb{Z}[\jj] \\
        \downarrow{\pi_{\ell \to k}}
            & & \downarrow{\pi_{\ell \to k}} \\
        \mathbb{Z}/M_k\mathbb{Z}
            & \xrightarrow{T_k}
            & \mathbb{Z}/M_k\mathbb{Z}[\jj]
    \end{array}
\]
commutes for every $k \leq \ell$.
D-holomorphy is the second condition of $\mathrm{HolFun}$
(Definition~\ref{def:holfun}, I.D47).
Therefore $T \in \mathrm{Nat}(\mathcal{F}, \mathcal{F}_{\jj}) \cap \mathrm{D\text{-}Hol}$.

$(\supseteq)$\;
Let $T \in \mathrm{Nat}(\mathcal{F}, \mathcal{F}_{\jj}) \cap \mathrm{D\text{-}Hol}$.
Naturality of $T$ gives the commuting square above,
which is exactly the tower coherence condition (I.D46).
D-holomorphy gives sector independence (I.P22).
Together these are the two conditions
of $\mathrm{HolFun}$ (I.D47).