%
\label{thm:distributivity}
For all presheaves $P, Q, R$ in $\mathcal{E}_\tau$:
\[
    \boxed{%
    P \times (Q \wedge R)
    \;\cong\;
    (P \times Q) \wedge (P \times R).}
\]
The categorical product distributes
over the categorical coproduct.

We verify the isomorphism pointwise.
At each object $X$ in $\mathrm{Cat}_\tau$:
\begin{align*}
    \bigl(P \times (Q \wedge R)\bigr)(X)
    &= P(X) \times \bigl(Q(X) \vee R(X)\bigr) \\
    &= \bigl(P(X) \times Q(X)\bigr)
       \vee \bigl(P(X) \times R(X)\bigr) \\
    &= \bigl((P \times Q) \wedge (P \times R)\bigr)(X).
\end{align*}
The second equality is the set-theoretic distributive law
$A \times (B \cup C) = (A \times B) \cup (A \times C)$.
Naturality holds because
both product and coproduct are defined pointwise,
so restriction maps commute with the isomorphism at each stage.
By commutativity of~$\times$,
right-distributivity
$(Q \wedge R) \times P
\cong (Q \times P) \wedge (R \times P)$
follows immediately.