= Solution
Write $\widehat{\mathcal C}=[\mathcal C^{\mathrm{op}},\mathbf{Set}]$ and similarly for $\mathcal D$. The <geometric morphism induced by a functor> has
$$
\boxed{f^*P=P\circ F^{\mathrm{op}},\qquad f_*Q=\operatorname{Ran}_{F^{\mathrm{op}}}Q.}
$$
Precomposition preserves all pointwise limits and colimits, so in particular it preserves <finite limits>. The <Right Kan extension> exists because the categories are small and sets have all small limits; its universal property gives $f^*\dashv f_*$. Thus these functors define a <geometric morphism> $\widehat{\mathcal C}\to\widehat{\mathcal D}$.
There is also $f_!=\operatorname{Lan}_{F^{\mathrm{op}}}$, a <left Kan extension>, with $f_!\dashv f^*$. The <Yoneda lemma> identifies $f_!(yC)\cong y(FC)$, since for every $P$,
$$
\operatorname{Hom}(f_!yC,P)\cong\operatorname{Hom}(yC,f^*P)\cong P(FC)\cong\operatorname{Hom}(y(FC),P).
$$
This is the <representable> calculation used in the next parts.
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