- I find that the restriction and pullback of representable etale sheaf along closed immersion are very confusing. I think they are different in general, I hope some experts can confirm my understanding about this.

For simplicity, let's work over a trait $S$, with $s\xrightarrow{i}S\xleftarrow{j}\eta$ the special point and the generic point. Let $G$ be an $S$-scheme, considered as a sheaf on the etale site $(\mathrm{Sch}/S)_{\mathrm{et}}$. Now we have two sheaves associated to $i$.

(1) the restriction of the sheaf $G$ to the site $(\mathrm{Sch}/s)_{\mathrm{et}}$;

(2) the pullback $i^*G$ along the closed immersion $i$

Apparently, the first sheaf is represented by the scheme $G_s=G\times_{S}s$. How about the second one? It seems to me that the second one is represented by $G_s$ if for any scheme $V_s$ over $s$, there exists an 'canonical' ('universal') lifting $V$ over $S$ such that $V\times_{S}s=V_s$ and $Hom_S(V,G)\cong Hom_s(V_s,G_s)$. This rarely happenes.

2. I think a closed subscheme usually doesn't give rise to subsheaf. I used to be very confused with this.