%
\label{thm:grothendieck-topos}
$\mathrm{PSh}(\mathrm{Cat}_\tau)$
is a \textbf{Grothendieck topos}.

By Giraud's theorem \cite{MacLaneMoerdijk1992},
the presheaf category
$[\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]$
on any small category $\mathcal{C}$
is a Grothendieck topos.
$\mathrm{Cat}_\tau$ is small:
\begin{itemize}
    \item \textbf{Objects:}
          $\Obj(\mathrm{Cat}_\tau) = \tau\text{-Idx}
          \cong \mathbb{N}$ (a set).
    \item \textbf{Morphisms:}
          $\mathrm{Mor}(\mathrm{Cat}_\tau)
          = \{(X, Y) : X \mid Y\}
          \subseteq \tau\text{-Idx} \times \tau\text{-Idx}$
          (a set).
\end{itemize}
Explicitly, the four Giraud axioms hold:
\begin{enumerate}
    \item All small colimits exist
          (computed pointwise in $\mathbf{Set}$).
    \item The representables $\{y(X)\}_{X \in \mathrm{Cat}_\tau}$
          form a small generating set.
    \item Coproducts are disjoint
          (inherited from $\mathbf{Set}$).
    \item Equivalence relations are effective
          (inherited from $\mathbf{Set}$).
\end{enumerate}