= Solution
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 gives
$$
\alpha_d=Q(i)\,\alpha_{Tc}\,P(r).
$$
Consequently transformations agreeing at all $Tc$ agree at every retract of such an object. If every $d$ is such a retract, restriction $f^*$ is faithful, and the <geometric morphism> is surjective.
Conversely suppose $f^*$ 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 $f^*S_d=f^*yd$. 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>:
$$
\boxed{f\text{ surjective}\Longleftrightarrow
\text{every }d\in\mathbb D\text{ is a retract of some }Tc.}
$$
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