%
\label{def:finite-limits}
The category $\mathrm{Cat}_\tau$
(Definition~\ref{def:cat-tau}, I.D51)
admits the following finite limit constructions:
\begin{enumerate}
    \item \textbf{Terminal object.}
          The object $\mathbf{1}_\tau := \underline{1}$
          (the multiplicative identity in $\tau$-Idx)
          is terminal:
          for every object $X$ in $\mathrm{Cat}_\tau$,
          there exists a unique arrow
          $!_X : X \to \mathbf{1}_\tau$.
    \item \textbf{Binary products.}
          For objects $X, Y$ in $\mathrm{Cat}_\tau$,
          the product is:
          \[
              \boxed{%
              X \times_\tau Y
              \;:=\;
              X \cdot Y,}
          \]
          where $\cdot$ denotes internal multiplication
          on $\tau$-Idx
          (Chapter~\ref{ch:swap-add-mul}).
          The projections $\pi_1 : X \cdot Y \to X$
          and $\pi_2 : X \cdot Y \to Y$
          are the $\tau$-holomorphic programs
          that extract the respective factor
          via the NF address encoding
          (Definition~\ref{def:nf-encoding}, I.D16).
    \item \textbf{Equalizers.}
          Since $\mathrm{Cat}_\tau$ is thin
          (at most one arrow between any two objects),
          parallel arrows $f, g : X \rightrightarrows Y$
          force $f = g$.
          The equalizer is therefore $X$ itself.
    \item \textbf{Pullbacks.}
          For arrows
          $f : X \to Z$ and $g : Y \to Z$,
          the pullback reduces to $\gcd(X, Y)$
          in the thin category $\mathrm{Cat}_\tau$.
\end{enumerate}