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\label{def:holend-tau}
The \textbf{holomorphic endomorphism category}
$\mathrm{HolEnd}_\tau$ is defined by:
\begin{enumerate}
    \item\textbf{Objects:}
          $\tau$-admissible points
          (II.D02,
          Chapter~\ref{ch:tau-admissible-points}).
    \item\textbf{Morphisms:}
          For objects $x, y$:
          $\mathrm{Hom}_{\mathrm{HolEnd}_\tau}(x, y)
          := \{$~$\omega$-germ transformers
          from $x$ to $y$$\}$.
    \item\textbf{Composition:}
          Stagewise composition
          (Definition~\ref{def:composition}, II.D39).
    \item\textbf{Identity:}
          The constant family
          $\mathrm{id}_x = \{\mathrm{id}_k\}$
          (Definition~\ref{def:identity-map}, II.D40).
\end{enumerate}
The category axioms hold:
\begin{itemize}
    \item \textbf{Associativity}:
          Theorem~\ref{thm:associativity} (II.T29).
    \item \textbf{Left identity}:
          $\mathrm{id} \circ f = f$
          (Proposition~\ref{prop:ch33-identity-laws}).
    \item \textbf{Right identity}:
          $f \circ \mathrm{id} = f$
          (Proposition~\ref{prop:ch33-identity-laws}).
\end{itemize}