Equivalence of categories between sheaves on $X$ and sheaves on a given base $B$ of $X$. In Remark 2.7.2 of Vakil's notes, he remarks that sheaves on $X$ and sheaves on a base $B$ are equivalent categories. Let the two functors be ext and res, where ext extends a sheaf $F$ on $B$ to a sheave $\mathscr{F}$ on $X$, and res restricts a sheave $\mathscr{F}$ on $X$ to a sheave $F$ on $B$. I understand the equivalence of categories. I don't understand the question "Why aren't their compositions the identity" at the end of the remark. Isn't the restriction of extension of $F$ on $B$ itself and the extension of restriction of $\mathscr{F}$ on $X$ itself? Thank you for your help!

Start with a sheaf $\mathscr F$ on $X$, and then take $F:=\text{res }\mathscr F$ to get a sheaf on the base $B$. We'd like to extend this sheaf back to all of $X$ and see what happens. Well, now you should go look how this extension is constructed in Vakil. It is a large construction using classes of "compatible germs of $F$", so it is certainly not **equal** to the original sheaf $\mathscr F$. The two are **isomorphic** as sheaves on $X$, however.