Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 75 / 5 / i

Precomposition definesIt preserves finite limits because presheaf limits are pointwise. Its right adjoint is a Right Kan extension; explicitly, for $H\in \hat{\mathbb{C}}$,Precomposition in $d$ supplies the restriction maps. To verify the adjunction, a map $P\circ T^{\mathrm{op}}\to H$ 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 $f^{\ast }⊣f_{\ast }$ and $f^{\ast }$ is left exact, giving the required geometric morphism