I am going through SGA 1.
Grothendieck shows in Exposé 1 the following thing:
Let $S$ be a scheme, $X,Y$ two $S$-schemes, and let $S_0\to S$ be a closed immersion that is topological homemorphism. Let $X_0=X\times _SS_0$ and $Y_0=Y\times _SY_0$. We suppose $X$ étale over $S$. Then the natural map $$Hom_S(Y,X)\to Hom_{S_0}(Y_0,X_0)$$ is bijective.
The first step is clear, one can assume that $Y=S$. Then he claims that because of topological description of sections $X\to Y$ one has the bijection. However, in the topological description, there is no need for morphism to be étale, required is only unramified, however it is needed that $X\to Y$ is separated (to get the section closed immersion).
Here is the statement (5.3): Let $X\to Y$ be separated, unramified, $Y$ connected. Then sections of $X\to Y$ are in the correspondance with the connected components of $X$ such that the restriction morphism is an isomorphsim.
The proof is: section of unramified morphism is an étale morphism (4.8), section of separated morphism is closed immrersion, now we use closed immersion+étale=open immersion(5.2)
Can someone explain the missing step?
One indeed doesn't need $X$ to be étale over $S$, unramified is enough for functor $\times _SS_0$ to be fully faitful. However, if one wants to have equivalence of categories, one needs étalness (8.3)
The condition on separatedness is needed, it is mentioned both in the remark after the statement and in (8.3).