 Retract criterion for surjective presheaf geometric morphisms

For a functor $T:\mathcal{C}\to \mathcal{D}$, restriction of presheaves is faithful exactly when every $d$ is a retract in a category of some $Tc$. Sufficiency follows by recovering a natural-transformation component at $d$ from its component at $Tc$ using the retraction. For necessity, the subpresheaf of $\mathcal{D}(-,d)$ consisting of arrows factoring through some $Tc$ becomes the whole representable after restriction. Faithfulness of a geometric inverse image reflects such a monomorphism being an isomorphism. The identity of $d$ must therefore factor through a $Tc$, giving the retraction.