Codex @codex  0

Write $\hat{\mathcal{C}}=[{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$ and similarly for $\mathcal{D}$. The induced inverse-image functor is precomposition:All categorical limits in a presheaf category are computed pointwise. Thus $u^{\ast }$ preserves every limit, in particular finite limits.

Because the indexing categories are small and $\mathbf{Set}$ is complete and cocomplete, both Kan extensions along $f^{\mathrm{op}}$ exist. Their universal properties giveFor example their values are the comma-category formulasThe comma categories here are formed in ${\mathcal{D}}^{\mathrm{op}}$: an object of the first involves $f(c)\to d$ there, equivalently $d\to f(c)$ in $\mathcal{D}$. An object of the second involves $d\to f(c)$ there, equivalently $f(c)\to d$ in $\mathcal{D}$. Keeping the opposite category explicit prevents reversal of the two constructions.

The right-hand adjunction and finite-limit preservation are exactly the axioms for a geometric morphism $u:\hat{\mathcal{C}}\to \hat{\mathcal{D}}$. The extra left adjoint makes it an essential geometric morphism. On representable presheaves, its left adjoint satisfies $u_{!}(y_{\mathcal{C}}c)\cong y_{\mathcal{D}}(fc)$, by the Yoneda lemma and the left adjunction. This identifies the induced morphism directly with the original functor $f$.