what is restriction of a sheaf?

Question

I have a question about sheaf theory. Thank you.

In Hartshorne's book, the restriction of a sheaf is defined by using an inverse image sheaf, that is:

Let $X$ be a topological space, $U$ an open in $X$, $i:U\to X$ an inclusion and $\mathcal{F}$ be a sheaf on $X$. Then, the restriction of $\mathcal{F}$ to $U$ is defined by $\mathcal{F}|_U:=i^{-1}\mathcal{F}$.

However, according to some texts (for example, Ulrich Görtz | Torsten Wedhorn, Algebraic Geometry I), $\mathcal{F}|_U(V):=\mathcal{F}(V)$.

So, $i^{-1}\mathcal{F}(V)=\mathcal{F}(V)$? Why?