%
\label{def:holmap}
A \textbf{$\tau$-holomorphic map} is a triple
$(T, \mathrm{src}, \mathrm{tgt})$ where:
\begin{enumerate}
    \item $T \in \mathrm{HolFun}$
          is a $\tau$-holomorphic function
          (Definition~\ref{def:holfun}, I.D47).
    \item $\mathrm{src}$ is an object of $\tau$
          (the source),
          determining the rooting of the input omega-tails.
    \item $\mathrm{tgt}$ is an object of $\tau$
          (the target),
          determining the compatibility condition
          on the output split-complex values:
          $T$ maps omega-tails rooted at $\mathrm{src}$
          to split-complex values compatible with $\mathrm{tgt}$.
\end{enumerate}
The type of all $\tau$-holomorphic maps is denoted:
\[
    \boxed{%
    \mathrm{HolMap}
    \;:=\;
    \bigl\{
        (T, \mathrm{src}, \mathrm{tgt})
        \;\big|\;
        T \in \mathrm{HolFun},\;
        \mathrm{src}, \mathrm{tgt} \in \mathrm{Ob}(\tau)
    \bigr\}.}
\]