%
\label{def:bi-monoidal}
The \textbf{bi-monoidal structure} on $\mathcal{E}_\tau$ is the triple:
\[
    \boxed{%
    (\mathcal{E}_\tau,\; \times,\; \wedge),}
\]
where:
\begin{enumerate}
    \item $(\mathcal{E}_\tau, \times, \mathbf{1})$
          is a monoidal category
          under the cartesian product
          (Chapter~\ref{ch:cartesian-product}),
          with terminal presheaf $\mathbf{1}$ as unit.
    \item $(\mathcal{E}_\tau, \wedge, \mathbf{0})$
          is a monoidal category
          under the coproduct
          (Definition~\ref{def:categorical-coproduct}, I.D62),
          with initial presheaf
          $\mathbf{0}$ ($\mathbf{0}(X) = \varnothing$) as unit.
    \item $\times$ distributes over~$\wedge$
          (Theorem~\ref{thm:distributivity}, I.T27).
    \item $\mathbf{0}$ annihilates under~$\times$:
          $P \times \mathbf{0} \cong \mathbf{0}$
          (since $P(X) \times \varnothing = \varnothing$).
\end{enumerate}
Both monoidal structures are symmetric
($P \times Q \cong Q \times P$,
$P \wedge Q \cong Q \wedge P$),
making $(\mathcal{E}_\tau, \times, \wedge)$
a \textbf{symmetric bi-monoidal category}.