In Görtz/Wedhorn it is stated that if $Z$ is a locally closed subset of a scheme $X$, then there exists a unique reduced subscheme $Z_{\operatorname{red}}$ with the same topological space as $Z$.
My questions: why is this unqie? (Basically I would like to understand the argument (one sentence) they give for this assertion.)
My thoughts: basically we could start with any (sub)scheme structure on $Z$ and apply the functor $\cdot_{\operatorname{red}}$ to obtain a reduced subscheme, but if we start with different scheme structures at the outset, why don't we end up with different reduced scheme structures? Görtz/Wedhorn refers to the Proposition before the statement, which says that $\cdot_{\operatorname{red}}$ is functorial, so my second thought was that if we assume we have a reduced scheme structure $Z_{\operatorname{red}}$ (where now the $_{\operatorname{red}}$ is rather a fixed expression than a functor I guess), we could apply the functor $\cdot_{\operatorname{red}}$ to the closed immersion $i:Z_{\operatorname{red}}\rightarrow X$ and obtain $(Z_{\operatorname{red}})_{\operatorname{red}}=Z_{\operatorname{red}}$, so that $$i=Z_{\operatorname{red}}\stackrel{i_{\operatorname{red}}}{\rightarrow}X_{\operatorname{red}}\hookrightarrow X,$$ and I thought maybe I could argue from here somewhere, but I have no clue how to go on. I'm grateful for any sort of help.
