A classical fibre is base change to a point

Points of a classical affine chart over $k$ are closed by the point/maximal-ideal correspondence in Classical affine algebraic sets correspond to radical ideals, and irreducible sets to prime ideals, so $\{s\}$ is closed in $S$. Regular maps are continuous by the chart convention and A classical morphism pulls Zariski closed sets back to closed sets. Thus $F=f^{-1}(s)$ is closed in $X$. Every point $x\in F$ has affine charts $W\subseteq X$ and $V\subseteq S$ around $x$ and $s$ with $f(W)\subseteq V$: shrink a source chart within the open preimage of $V$ to a principal affine chart.

On such charts put $J_W=k[W]f^{\ast }({\mathfrak{m}}_s)$ and $F_W=V_W(J_W)$. Its points are exactly $W\cap F$, because the coordinate functions of $V$ distinguish $s$ from every other point. The relative Nullstellensatz Classical affine algebraic sets correspond to radical ideals, and irreducible sets to prime ideals gives $I_W(F_W)=\sqrt{J_W}$, hence $k[F_W]=k[W]\mathrm{/}\sqrt{J_W}$. These reduced affine algebraic-set structures agree under restriction on overlaps. On a common principal-open refinement, localization commutes with radicals: $\sqrt{J_W}k[W{]}_h=\sqrt{J_{W}k[W{]}_h}$; both chart presentations therefore give the same reduced quotient and the same sheaf of locally regular functions on the closed locus. They glue to a classical prevariety structure on $F$, including the empty case. It is quasi-compact because $F$ is closed in the quasi-compact $X$. The inclusion $F\to X$ is regular and its composite with $f$ is constant at $s$.

Let $T$ be a classical prevariety with regular map $g:T\to X$ such that $f\circ g$ is constant at $s$. On affine charts of $T$ mapping into a chart $W$ of step 2.1, the pullback $g^{\ast }:k[W]\to k[T_0]$ kills $J_W$. The target coordinate ring is reduced, so it also kills $\sqrt{J_W}$. The affine morphism/coordinate-ring correspondence Classical affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms factors $g$ uniquely through $F_W$. These local factorizations agree on overlaps because they all have the same underlying map into the closed subspace $F$ and the same pullback of locally regular functions. They glue to a unique regular map $T\to F$. Thus $F$ has the pullback property of Base change of classical varieties when the pullback exists, so any previously constructed pullback is canonically isomorphic to it. If $X$ and $S$ are separated, the closed subspace $F\subseteq X$ is separated, proving the last clause.