Let $X$ be a scheme and $\mathcal{F}$ a presheaf on $X_{ét}$.

For each $x_{i}\in X$, pick a geometric point $\bar{x}_{i}$ over $x$ and denote by $i_{\bar{x}_{i}}:\text{Spec}(k_{i})_{\text{ét}}\rightarrow X_{ét}$ the morphism of sites induced by the geometric point $\bar{x}_{i}$ where $k_{i}$ is algebraically closed. Then we can define a sheaf on $X_{ét}$

$$\text{Esp}(\mathcal{F}):=\prod_{\bar{x}_{i}}(i_{\bar{x}})_{*}\mathcal{F}_{\bar{x}}.$$

What is the map $\mathcal{F}\rightarrow \text{Esp}(\mathcal{F})$? I assume you would send sections to their germs, however if $U\rightarrow X$ is étale then $$ (i_{\bar{x}})_{*}\mathcal{F}_{\bar{x}}(U)=\prod_{\text{Hom}_{X}(\bar{x},U)}\mathcal{F}_{\bar{x}}. $$ So would you have to send a section to the germ then compose with some kind diagonal embedding?