The Stacks project

Example 7.14.2. Let $f : X \to Y$ be a continuous map of topological spaces. Recall that we have sites $X_{Zar}$ and $Y_{Zar}$, see Example 7.6.4. Consider the functor $u : Y_{Zar} \to X_{Zar}$, $V \mapsto f^{-1}(V)$. This functor is clearly continuous because inverse images of open coverings are open coverings. (Actually, this depends on how you chose sets of coverings for $X_{Zar}$ and $Y_{Zar}$. But in any case the functor is quasi-continuous, see Remark 7.13.6.) It is easy to verify that the functor $u^ s$ equals the usual pushforward functor $f_*$ from topology. Hence, since $u_ s$ is an adjoint and since the usual topological pullback functor $f^{-1}$ is an adjoint as well, we get a canonical isomorphism $f^{-1} \cong u_ s$. Since $f^{-1}$ is exact we deduce that $u_ s$ is exact. Hence $u$ defines a morphism of sites $f : X_{Zar} \to Y_{Zar}$, which we may denote $f$ as well since we've already seen the functors $u_ s, u^ s$ agree with their usual notions anyway.