Assume we are given an etale sheaf $\mathcal{F}$ on the big etale site of schemes over $S$. Let $X$ be a $S$ scheme with the structure morphism $j$ and let $i:\eta_X\hookrightarrow X$ be its generic point. Is it true that $(j\circ i)^*\mathcal{F}\simeq \mathcal{F}|_{\eta_X}$. The pullback means looking at $(j\circ i)^*\mathcal{F}$ as a sheaf on the small etale site of the generic point (after pullback in the big etale site). The sheaf ${F}|_{\eta_X}$ means restricting the values of $\mathcal{F}$ only to $\eta_X$ schemes (since $\eta_X$ schemes are $S$ schemes $\mathcal{F}$ has already some values defined for them.) and looking at it as a sheaf on the small etale site of $\eta_X$.
Generally pullback and restriction do not coincide unless that morphism is of finite type, I was wondering whether in the special case of the generic point or localization at a prime it is true.