= Solution
The canonical <geometric morphism> $\widehat{\mathcal C}\to\mathbf{Set}$ has inverse image the constant-presheaf functor $\Delta$ and direct image the <global sections functor>
$$
\Gamma(P)=\operatorname{Hom}(1,P),\qquad\Delta\dashv\Gamma.
$$
If the <presheaf topos> is a <local topos>, $\Gamma$ is also the inverse image of a <geometric morphism> $g:\mathbf{Set}\to\widehat{\mathcal C}$. This morphism has an extra <left adjoint> $\Delta$. Apply part (iii) with source category $1$ and target category $\mathcal C$, using its idempotent-splitting hypothesis. Then $g$ is induced by a functor $1\to\mathcal C$, choosing an object $C_0$, and $\Gamma$ is naturally evaluation at $C_0$.
Since evaluation at $C_0$ is $\operatorname{Hom}(yC_0,-)$, this says $\operatorname{Hom}(1,-)\cong\operatorname{Hom}(yC_0,-)$. Uniqueness of representing objects gives $yC_0\cong1$. Thus $\mathcal C(C,C_0)$ is a singleton for every $C$: $C_0$ is terminal.
Conversely, if $C_0$ is terminal, $yC_0=1$ and $\Gamma(P)=P(C_0)$. Evaluation at that object preserves <finite limits> and has a right Kan extension as <right adjoint>, so is an inverse image functor. Therefore, under the permitted idempotent-completeness assumption,
$$
\boxed{\widehat{\mathcal C}\text{ is local if and only if }\mathcal C\text{ has a terminal object}.}
$$
Back to article page
