%
\label{def:holomorphic-presheaf}
The \textbf{holomorphic presheaf} $\mathcal{O}_\tau$
on~$\tau^3$ assigns to each cylinder domain
$U \subseteq \tau^3$ the set
\[
    \boxed{%
    \mathcal{O}_\tau(U)
    \;:=\;
    \bigl\{\,
    f : U \to H_\tau
    \;\big|\;
    f \text{ is $\tau$-holomorphic on } U
    \,\bigr\},}
\]
where $\tau$-holomorphic means
that $f$ satisfies the five equivalent conditions
of the Mutual Determination Theorem
(Theorem~\ref{thm:mutual-determination}, II.T27)
restricted to~$U$.

For an inclusion of cylinder domains
$V \subseteq U$,
the \textbf{restriction map}
\[
    \rho_{U,V} : \mathcal{O}_\tau(U) \to \mathcal{O}_\tau(V),
    \qquad
    \rho_{U,V}(f) := f\big|_V,
\]
is the ordinary restriction of functions.

\medskip\noindent
The presheaf axioms are immediate:
\begin{enumerate}
    \item[\textup{(P1)}]
          $\rho_{U,U} = \mathrm{id}$
          (restriction to the same domain is the identity).
    \item[\textup{(P2)}]
          $\rho_{V,W} \circ \rho_{U,V} = \rho_{U,W}$
          for $W \subseteq V \subseteq U$
          (restriction is transitive).
\end{enumerate}
Both are trivially satisfied
by function restriction.