It's an exercise and some text from Vakil’s FOAG. I quote the related context as the following screenshot(sorry for posting it as a picture)
I am confused about the difference between $Exercise$ $2.5.A$ and $Theorem$ 2.5.1.3 It seems $Theorem$ $2.5.1$ formalizes and answers $Exercise$ $2.5.A$ perfectly. Is it any difference between these two I haven’t noticed?
Related question maybe Recovering a sheaf from base of topology and Recovering sheaf from sheaf on base.. Reading this two posts I still cannot see what does $Exercise$ $2.5.A$ differ from $Theorem$ $2.5.1.$
Thank you very much.
You're right they're very related. The difference really is that in the setup of Theorem 2.5.1 there is no sheaf on $X$, you are starting with a sheaf on the base and showing that you can define a corresponding sheaf on $X$. On the other hand, for the exercise you are starting with a sheaf on $X$ in the first place, forgetting all but the sheaf info "on the base" and showing you still have enough information to recover your original sheaf on $X$. Because the construction in the theorem is indeed the same thing you'd do to recover your sheaf in the exercise, you could phrase this in other terms as the verification that the composition
$$\text{(sheaf on $X$) $\to$ (sheaf on base) $\to$ (sheaf on $X$)}$$
is the identity, where the second map is that which Theorem 2.5.1 defines. Thus the exercise is part of the verification that you actually have a bijection between sheaves on the base and sheaves on $X$.