= 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.
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