# Construction of Grassmann variety representing the Grassmann functor

I'm reading "The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn, and I am stuck with the construction of the Grassmann variety (Example 2.2.2). I'd appreciate any help, so here goes:

Let V be a be a finite dimensional vector space over a field $k$. Let $0 \leq r \leq dim(V)$. Define the Grassmann functor as $\underline{Grass}(V,r):(Sch/k)^o \rightarrow (Sets)$ associating to any $k$-scheme $S$ of finite type the set of all subsheaves $K \subset \mathcal{O}_S \otimes V$ with locally free quotient $F = \mathcal{O}_S \otimes V/K$ of rank $r$.

For each $r$-dimensional linear subspace $W \subset V$ they consider a subfunctor $\mathcal{G}_W$ of $\underline{Grass}(V,r)$, associating to each $k$-scheme S those locally free quotients $F$ for which the composition $\mathcal{O}_S \otimes W \rightarrow \mathcal{O}_S \otimes V \rightarrow F$ is an isomorphism, giving a splitting of the inclusion $W \subset V$. Now they make the unclear jump to saying that from this one can conclude that $\mathcal{G}_W$ is represented by $\text{Spec}(S^*\text{Hom}(V,W)^{\vee})$, "corresponding to homomorphisms that split the inclusion map $W \subset V$". Now, if I am interpreting this correctly $\text{Spec}(S^*\text{Hom}(V,W)^{\vee})$ is an affine scheme over the vector space of polynomial functions on the vector space $\text{Hom}(V,W)$. But I don't see how an element of $\mathcal{G}_W(S)$ gives a map of schemes $S \rightarrow \text{Spec}(S^*\text{Hom}(V,W)^{\vee})$. Additionally, what is the reverse map, given such a morphism of schemes $S \rightarrow \text{Spec}(S^*\text{Hom}(V,W)^{\vee})$, how does one obtain a locally free quotient in $\mathcal{G}_W(S)$?

Using the isomorphism $F\cong \mathcal{O}_S\otimes W$ implies that an $S$-point of $\mathcal{G}_W$ gives a map $\mathcal{O}_S\otimes V \to \mathcal{O}_S\otimes W$ which splits the inclusion. This is the desired $S$-point of the scheme $\mathrm{Spec}(S^*\mathrm{Hom}(V,W)^{\vee})$, though we should note that it is not this whole space that represents $\mathcal{G}_W$ but just the affine subspace consisting of splittings.