I want to verify the sentence:
...can check pullback of the “space of sections” is the “space of sections” of the pullback
in Vakil's Exercise 2.7.E, the latest version.
Let $f:X\to Y$ be a continuous map and $\mathcal G$ be a (pre)sheaf over $Y$ and $G=\bigsqcup_q\mathcal G_q$ be the space of sections (espace etale) as defined in his 2.2.11.
I think the "space of sections of the pullback" is $F=\bigsqcup_p(f^{-1}\mathcal G)_p$, where $f^{-1}\mathcal G$ is the inverse image sheaf.
But I don't know what is the "pullback of the space of sections". I guess I must also prove that they are isomorphic as topological spaces, not just as sets.
If my interpretation is totally wrong, please ignore it and show me the correct interpretation. I only know the definition of the space of sections (espace etale).