Isn't $\operatorname{Hom}_S(T,S)$ just a singleton? This is Proposition 8.4 of Görtz/Wedhorn. It is stated that if $v:\mathscr{E}\rightarrow \mathscr{F}$ is a homomorphism of quasi-coherent $\mathcal{O}_S$-modules, and $\mathscr{F}$ is of finite type, then the functor $F: (\operatorname{Sch}/S)^{\operatorname{opp}}\rightarrow (\operatorname{Sets})$ defined by $F(T):=\{f\in \operatorname{Hom}_S(T,S)\mid f^*(v) \operatorname{ is surjective}\}$ is represented by an open subscheme of $S$.
Now it appears to me to be evident that $\operatorname{Hom}_S(T,S)$ is just the singleton consisting of the structur morphism of $T$. However, in this case the statement of the proposition would seem to be not really interesting, so I am sure that I am mistaken, which confuses me quite a bit. I would ge glad if somebody could tell me where my mistake is.
 A: The set $F(T)$ is a subsingleton. It is a singleton when $f^*(v)$ is surjective for the structure morphism $f : T \to S$, and otherwise empty. The Proposition is still interesting. It says that there is a largest open subscheme $S' \subseteq S$ such that $v|_{S'}$ is surjective. Of course, this is just the complement of the support of $\mathrm{coker}(v)$. The support is closed since $\mathrm{coker}(v)$ is of finite type.
A: Indeed, $\operatorname{Hom}_S(T,S)$ is a singleton, so $F(T)$ is always either empty or a singleton.  The proposition is still nontrivial, though: it tells you that the test of whether $F(T)$ is empty or not is given by an open subset of $S$ (and $F(T)$ being nonempty means that the pullback of $v$ to $T$ via the structure morphism is surjective).  That is, $F$ is represented by some open subscheme $U$ of $S$, which just means that $F(T)$ is nonempty iff the structure morphism $T\to S$ has image contained in $U$.  In other words, the schemes over $S$ on which $v$ pulls back to a surjection are exactly the ones whose structure morphisms have image contained in $U$.
One reason they might choose to state the proposition in this somewhat convoluted way is they may want to connect it with a more general statement, where for instance $v$ is not over the base scheme $S$ but instead some other scheme.
