Equalizers into separated schemes are closed

By [F1], with existence of fibre products, the fibre product $E$ of $(a,b):X\to Y{\times }_{S}Y$ and ${\mathrm{\Delta }}_{Y\mathrm{/}S}:Y\to Y{\times }_{S}Y$ exists, and for every scheme $T$ its $T$-points are the pairs $(t,c)$ with $t:T\to X$, $c:T\to Y$ and $(at,bt)={\mathrm{\Delta }}_{Y\mathrm{/}S}c$.

By [F3], separatedness of $Y\to S$ says that ${\mathrm{\Delta }}_{Y\mathrm{/}S}$ is a closed immersion, hence a monomorphism by [F5]; and by [F2] the composites of ${\mathrm{\Delta }}_{Y\mathrm{/}S}$ with the two projections are the identity, so $(at,bt)={\mathrm{\Delta }}_{Y\mathrm{/}S}c$ forces $at={\mathrm{pr}}_1{\mathrm{\Delta }}_{Y\mathrm{/}S}c=c={\mathrm{pr}}_2{\mathrm{\Delta }}_{Y\mathrm{/}S}c=bt$.

Consequently the $T$-points of $E$ are exactly the morphisms $t:T\to X$ with $at=bt$: given such $t$, the pair $(t,at)$ satisfies $(at,bt)=(at,at)={\mathrm{\Delta }}_{Y\mathrm{/}S}(at)$ by [F2]; conversely for a point $(t,c)$ of $E$ the equation $at=bt$ holds by step 1.2 and then $c=at$, so the second component is determined by $t$. This description is natural in $T$ and refers to no reducedness of $T$.

The projection $E\to X$ is the base change of ${\mathrm{\Delta }}_{Y\mathrm{/}S}$ along $(a,b)$ by the universal property of [F1]; as ${\mathrm{\Delta }}_{Y\mathrm{/}S}$ is a closed immersion by step 1.2, [F4] makes $E\to X$ a closed immersion.

Steps 1.1, 2.1 and 2.2 exhibit $E$ as a closed subscheme of $X$ whose $T$-points are precisely the $t:T\to X$ with $at=bt$, for every scheme $T$. This is the equalizer of $a$ and $b$, so the equalizer exists as a closed subscheme of $X$ and represents agreement of the two morphisms.