I am still studying Nitsure's Construction of Hilbert and Quot schemes, and I am yet again stuck, this time on page 25. I understand what lemma 5.4 says, but I do not get its proof. Let me sketch the outline:
$X\longrightarrow S$ is a projective morphism where $S$ is Noetherian, and $\psi:E\longrightarrow F$ a surjective morphism between coherent $X$-sheaves. They construct a closed subscheme $T'\subseteq T$ as follows:
[...] This is satisfied by taking $T'$ to be the vanishing scheme for the composite homomorphism $\ker(\phi)\hookrightarrow E\longrightarrow F$ of coherent sheaves over $X_{T}$.
To me, the $T'$ thus constructed is a subset of $X_{T}$, not of $T$.
EDIT: Also, why exactly does this lemma prove that the representability of $\mathfrak{Quot}^{\phi,L}_{G/X/S}$ implies the representability of $\mathfrak{Quot}^{\phi,L}_{E/X/S}$?