%
\label{def:categorical-coproduct}
For presheaves
$P, Q : \mathrm{Cat}_\tau^{\mathrm{op}} \to \mathrm{Set}$
in $\mathcal{E}_\tau$,
the \textbf{categorical coproduct}
$P \wedge Q$ is the presheaf defined pointwise by:
\[
    \boxed{%
    (P \wedge Q)(X)
    \;:=\;
    P(X) \vee Q(X)}
\]
for each object $X$ in $\mathrm{Cat}_\tau$,
where $\vee$ is Boolean disjunction
on the membership predicate:
$x$ belongs to $(P \wedge Q)(X)$
if and only if $x \in P(X)$ or $x \in Q(X)$ (or both).
On morphisms, the restriction maps act componentwise:
$(P \wedge Q)(f) := P(f) \vee Q(f)$.

The \textbf{coprojection morphisms} are the natural inclusions:
\[
    \iota_P : P \hookrightarrow P \wedge Q,
    \qquad
    \iota_Q : Q \hookrightarrow P \wedge Q.
\]
The coproduct satisfies the universal property:
for every presheaf $R$ and morphisms
$\alpha : P \to R$, $\beta : Q \to R$,
there exists a unique copairing
$[\alpha, \beta] : P \wedge Q \to R$
with $[\alpha, \beta] \circ \iota_P = \alpha$
and $[\alpha, \beta] \circ \iota_Q = \beta$.
Uniqueness follows from the thinness of $\mathrm{Cat}_\tau$
(Proposition~\ref{prop:thin-category},
Chapter~\ref{ch:earned-arrows}).