%
\label{def:cartesian-monoidal}
The \textbf{cartesian monoidal structure} on $\mathcal{E}_\tau$ is:
\[
    \boxed{%
    (\mathcal{E}_\tau,\; \times,\; \mathbf{1}),}
\]
where $\times$ is the categorical product bifunctor
(Definition~\ref{def:categorical-product}, I.D60)
and $\mathbf{1}$ is the terminal presheaf.
The coherence data:
\begin{enumerate}
    \item \textbf{Associator.}
          $\alpha_{P,Q,R} : (P \times Q) \times R
          \xrightarrow{\sim} P \times (Q \times R)$,
          $\alpha_X\bigl((s, t), u\bigr) = \bigl(s, (t, u)\bigr)$.
    \item \textbf{Unitors.}
          $\lambda_P : \mathbf{1} \times P \xrightarrow{\sim} P$,
          $\lambda_X(*, s) = s$;
          $\rho_P : P \times \mathbf{1} \xrightarrow{\sim} P$,
          $\rho_X(s, *) = s$.
    \item \textbf{Symmetry.}
          $\sigma_{P,Q} : P \times Q
          \xrightarrow{\sim} Q \times P$,
          $\sigma_X(s, t) = (t, s)$.
\end{enumerate}
These satisfy the pentagon and triangle coherence conditions.
All reduce pointwise to the corresponding identities
for the cartesian product of sets.