Etale neighbourhoods and elementary etale neighbourhoods of a point

Let $S$ be a scheme (Schemes) and let $s\in S$ be a point, with residue field $\kappa (s)$ (The residue field at a point of an affine scheme).

An 'etale neighbourhood of $(S,s)$ is a pair $(U,u)$ consisting of a scheme $U$, a morphism of schemes $\phi :U\to S$ that is 'etale (Étale morphism of schemes) and a point $u\in U$ with $\phi (u)=s$. One writes $\phi :(U,u)\to (S,s)$. A morphism of 'etale neighbourhoods $f:(V,v)\to (U,u)$ of $(S,s)$ is a morphism of $S$-schemes (Schemes and morphisms over a base) $f:V\to U$ with $f(v)=u$.

The 'etale neighbourhood $(U,u)\to (S,s)$ is elementary if the canonical map of residue fields $\kappa (s)\to \kappa (u)$ is an isomorphism, that is, if $\kappa (u)=\kappa (s)$.

Three conventions belong to the definition. First, no affineness is required of $U$; in the arguments on this page the neighbourhoods produced are affine when this is useful, and the definition above is the general one of Stacks Definition 37.35.1. Second, the point $u$ need not be closed in $U$, and the morphism $U\to S$ is required to be 'etale as a morphism, not merely 'etale at the point $u$; a morphism that is 'etale only on an open neighbourhood of $u$ gives an 'etale neighbourhood after replacing $U$ by that neighbourhood. Third, if $U\subseteq S$ is an open neighbourhood of $s$, then $(U,s)\to (S,s)$ is an 'etale neighbourhood, and it is elementary since the residue field at $s$ is unchanged; the composition of two 'etale neighbourhoods of $(S,s)$ is again one, and the composition of elementary ones is elementary because both residue field maps are isomorphisms.