I'm reading online lectures notes on sheaves, and I'm confused about the meaning of the equation $F|_U=0$ below. $F|_U$ is defined as the inverse image sheaf $f^\ast F$ along the inclusion of some subspace $f:U\hookrightarrow X$. What does it mean for a sheaf to be zero? Does it mean the sheaf always returns the zero object? Would it mean it always returns the empty set if the sheaf is $\mathsf{Set}$-valued?
Just for context, the proposition afterwards is $\mathrm{supp}(F)= \overline{\left\{ x\in X:F_x\neq 0 \right\}}$.
If you're working with sheaves in a category that has a zero object $0$ (for example abelian groups, rings, vector spaces ...), then the zero sheaf on $X$ is the sheaf $Z$ such that $Z(V) = 0$ for all open $V\subset X$.
So for an open subset $U\subset X$, you have $F\lvert_U = 0$ if and only if $F(V) = 0$ for all open $V\subset U$. In terms of étalé spaces, that means all stalks $F_x$ for $x\in U$ are $0$.