%
\label{def:hom-object}
For $A, B \in \mathrm{Obj}(\tau)$,
the \textbf{Hom object} $[A,B]$ is defined as the inverse limit
\[
    \boxed{%
    [A,B]
    \;:=\;
    \varprojlim_{k}
    \,\Hom_{\tau_k}(A_k, B_k),}
\]
where the connecting maps
$r_{k+1,k} : \Hom_{\tau_{k+1}}(A_{k+1}, B_{k+1})
\to \Hom_{\tau_k}(A_k, B_k)$
are the natural restriction maps:
\[
    r_{k+1,k}(f_{k+1})
    \;:=\;
    f_{k+1} \bmod P_k,
\]
i.e., the reduction of a stage-$(k+1)$ morphism
to its stage-$k$ shadow
via the natural projection
$\mathbb{Z}/P_{k+1}\mathbb{Z}
\to \mathbb{Z}/P_k\mathbb{Z}$.

\medskip\noindent
The Hom object carries the following structure:
\begin{enumerate}
    \item[\textup{(H1)}]
          \textbf{NF-addressability.}
          An element of $[A,B]$
          is a tower-coherent sequence $(f_k)_{k \geq 1}$
          of finite morphisms.
          Each $f_k$ has an NF-address at stage~$k$.
          The coherent family
          $(f_k)_{k \geq 1}$ determines
          a profinite NF-address for $[A,B]$.

    \item[\textup{(H2)}]
          \textbf{$H_\tau$-valued.}
          Each $f_k$ is valued in $H_\tau$.
          The limit inherits $H_\tau$-valued structure
          by pointwise inverse limit.

    \item[\textup{(H3)}]
          \textbf{Tower coherence.}
          The connecting maps $r_{k+1,k}$
          ensure that the sequence $(f_k)$ satisfies
          $f_k \equiv f_{k+1} \pmod{P_k}$---the
          same tower coherence condition
          that defines $\tau$-holomorphic maps.

    \item[\textup{(H4)}]
          \textbf{Ultrametric structure.}
          The inverse limit inherits the ultrametric topology
          from the tower
          (Chapter~\ref{ch:ultrametric-depth}, II.D13):
          two elements of $[A,B]$ are close
          if they agree at deep stages.
\end{enumerate}