I am reading in the paper of Nitsure (link: http://arxiv.org/pdf/math/0504590v1.pdf) about the construction of the Quot-scheme $\mathrm{Quot}_{E/X/S}^{\Phi,L}$. After Lemma 5.4. they reduce to the case where $E$ is the pullback of a vector bundle on $S$ and $X=\mathbb{P}(V)$ is some projective space. I understand the proof of Lemma 5.4., and the lemma implies that we can assume $E$ as above. But how does the reduction to $X=\mathbb{P}(V)$ work?
Is it true, that there is an isomorphism of functors $\mathrm{Quot}_{E/X/S}^{\Phi,L} \rightarrow \mathrm{Quot}_{E'/\mathbb{P}(V)/S}^{\Phi,L}$ when $X \hookrightarrow \mathbb{P}(V)$ is a closed subscheme? At least there's a morphism from the right to the left by taking pullbacks. However, I don't see how an inverse would look like. Or is there a closed embedding $\mathrm{Quot}_{E/X/S}^{\Phi,L} \rightarrow \mathrm{Quot}_{E'/\mathbb{P}(V)/S}^{\Phi,L}$ which would suffice as well?
Here is some other idea: a quotient of $E$ on $X$ is the same as a quotient of the pushforward $i_{\ast}E$ on $\mathbb{P}(V)$. But is it guaranteed, that $i_{\ast}E$ is still a quotient of some $\pi^{\ast}W$?