Question about notation for Čech cohomology and direct image of sheaves in Hartshorne

Question

Oh page 220 of Hartshorne, he defines what he describes as a "sheafified" version of the Čech complex. Let $X$ be a topological space with cover $\mathfrak{U}$ and sheaf of abelian groups $\mathcal{F}$. Let $V \subseteq X$ be an open subset of $X$ with inclusion map $j : V \hookrightarrow X$. He then defines a complex $\mathscr{C}^{\bullet}(\mathfrak{U}, \mathcal{F})$ of sheaves on $X$ with $$ \mathscr{C}^{p}(\mathfrak{U}, \mathcal{F}) = \prod_{i_{0} < i_{1} < \cdots < i_{p}} j_{*}\left(\mathcal{F}|_{U_{i_{0}i_{1} \cdots i_{p}}} \right) $$ My confusion is about what this direct image functor $j_{*}$ means. Is $\mathcal{F}$ supposed to be further restricted down to $V$? There isn't even any assumption of irreducibility so there's no reason to think a finite collection of the $U_{i}$ even intersect $V$. By definition $j_{*}$ is supposed to be a functor from the category of sheaves on $V$. How should I interpret $\mathcal{F}|_{U_{i_{0}i_{1} \cdots i_{p}}}$ as a sheaf on $V$?