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If $d$ is a retract in a category of $Tc$, write $i:d\to Tc$, $r:Tc\to d$ with $ri=1_d$. For a natural transformation $\alpha :P\to Q$ of categorical presheaves, naturality givesConsequently transformations agreeing at all $Tc$ agree at every retract of such an object. If every $d$ is such a retract, restriction $f^{\ast }$ is faithful, and the geometric morphism is surjective.

Conversely suppose $f^{\ast }$ is faithful. Inside the representable presheaf $yd=\mathbb{D}(-,d)$, define $S_d(e)$ to consist of arrows $e\to d$ that factor through some $Tc$. Precomposition preserves this condition, so $S_d$ is a subpresheaf. At $e=Tc$, every arrow factors through that very object, so $f^{\ast }{S}_d=f^{\ast }yd$. A faithful geometric inverse image reflects an inclusion becoming invertible, as proved in Question 4(iii). Hence $S_d=yd$, and $1_d\in S_d(d)$ factors as $d\to Tc\to d$. This is the required retraction. We have proved the retract criterion for surjective presheaf geometric morphisms: