Universal vanishing locus for a map into a flat projective family

Work over an affine open of $S$, and present $E$ by vector bundles $E_1\to E_0\to E\to 0$ that are sums of invertible twists. For $j=0,1$, the sheaf $E_j^{\vee }\otimes F$ is still base-flat. Let $K_j^{\bullet }$ be its complex from [F1], and define $Q_j=\mathrm{coker}((K_j^1{)}^{\vee }\to (K_j^0{)}^{\vee })$. Since finite projectives commute with dual tensor comparison, $\mathrm{Hom}(Q_j,B)=\ker (K_j^0\otimes B\to K_j^1\otimes B)={\mathrm{Hom}}_{X_B}(E_{j,B},F_B)$ naturally for every base algebra $B$.

The presentation induces a natural transformation between these Hom functors, hence a map $Q_1\to Q_0$; let $Q$ be its cokernel. Left exactness of Hom, also after every pullback of the presentation, identifies ${\mathrm{Hom}}_{X_B}(E_B,F_B)$ with $\mathrm{Hom}(Q,B)$. Thus the Hom functor is the affine linear scheme $\mathrm{Spec}\mathrm{Sym}Q$. The map $u$ defines a section of this scheme, and the inverse image of its zero section is cut out by the image of $Q\to {\mathcal{O}}_S$ associated to $u$. This is a closed subscheme with the claimed property. The local constructions agree by that property and glue over $S$.