Fibres after base change

Given: The objects, hypotheses and conventions in the statement above.

For a morphism $f:X\to S$ and any point $s\in S$, its scheme-theoretic fibre is $X_s=X{\times }_S\mathrm{Spec}\kappa (s),$ viewed as a $\kappa (s)$-scheme. The map $\mathrm{Spec}\kappa (s)\to S$ is the canonical residue-field point from lem-field-valued-points-of-schemes, and the product is base change as in def-base-change-morphism-schemes. The point $s$ need not be closed. A fibre over a generic point is called a generic fibre. Empty fibres are allowed. (Scheme-theoretic fibre)

For $S^{\mathrm{\prime }\mathrm{\prime }}\stackrel{k}{\to }{S}^{\mathrm{\prime }}\stackrel{h}{\to }S$ and an $S$-scheme $X$, there is a canonical isomorphism $(X{\times }_S{S}^{\mathrm{\prime }}){\times }_{S^{\mathrm{\prime }}}{S}^{\mathrm{\prime }\mathrm{\prime }}\cong X{\times }_S{S}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{.}$ It is functorial in $X$ and compatible with the induced maps of $S$-schemes. (Iterated base change)

For every field $K$ and scheme $X$, morphisms $\mathrm{Spec}K\to X$ correspond bijectively to pairs $(x,\iota )$ with $x\in X$ and a field embedding $\iota :\kappa (x)\to K$. The identity embedding gives a canonical morphism $\mathrm{Spec}\kappa (x)\to X$, compatible with all scheme morphisms. More generally, for a nonzero local ring $(R,\mathfrak{m})$, morphisms $\mathrm{Spec}R\to X$ correspond to pairs $(x,\phi )$ with a local homomorphism $\phi :{\mathcal{O}}_{X,x}\to R$. Assuming Choice, two field-valued points have the same image in $X$ if and only if they are dominated by a common field-valued point, by compatible embeddings of their fields into a third field. (Field-valued points and local-ring points)