The Stacks project

Example 7.22.5. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ and $v : \mathcal{D} \to \mathcal{C}$ be functors of the underlying categories such that $v$ is right adjoint to $u$. Lemma 7.22.4 shows that if $v$ is continuous, then $u$ is cocontinous. Conversely, if $u$ is cocontinuous, then we can't conclude that $v$ is continuous. We will give an example of this phenomenon using the big étale and smooth sites of a scheme, but presumably there is an elementary example as well. Namely, consider a scheme $S$ and the sites $(\mathit{Sch}/S)_{\acute{e}tale}$ and $(\mathit{Sch}/S)_{smooth}$. We may assume these sites have the same underlying category, see Topologies, Remark 34.11.1. Let $u = v = \text{id}$. Then $u$ as a functor from $(\mathit{Sch}/S)_{\acute{e}tale}$ to $(\mathit{Sch}/S)_{smooth}$ is cocontinuous as every smooth covering of a scheme can be refined by an étale covering, see More on Morphisms, Lemma 37.38.7. Conversely, the functor $v$ from $(\mathit{Sch}/S)_{smooth}$ to $(\mathit{Sch}/S)_{\acute{e}tale}$ is not continuous as a smooth covering is not an étale covering in general.