= Solution
Precomposition defines
$$
f^*:[\mathbb D^{\rm op},\mathbf{Set}]\longrightarrow[\mathbb C^{\rm op},\mathbf{Set}],\qquad
P\longmapsto P\circ T^{\rm op}.
$$
It preserves <finite limits> because presheaf limits are pointwise. Its <right adjoint> is a <Right Kan extension>; explicitly, for $H\in\widehat{\mathbb C}$,
$$
(f_*H)(d)=\operatorname{Nat}_{\widehat{\mathbb C}}(\mathbb D(T-,d),H).
$$
Precomposition in $d$ supplies the restriction maps. To verify the <adjunction>, a map $P\circ T^{\rm op}\to H$ sends $x\in P(d)$ to the natural family taking $h:Tc\to d$ to the image in $H(c)$ of $P(h)x$. Conversely evaluate such a family at $d=Tc$, $h=1_{Tc}$; naturality makes the two constructions inverse. Thus $f^*\dashv f_*$ and $f^*$ is left exact, giving the required <geometric morphism>
$$
\boxed{\widehat{\mathbb C}\longrightarrow\widehat{\mathbb D}.}
$$
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