Sheafifcation for the étale site

Question

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?

Cross post:https://math.stackexchange.com/questions/4386913/sheaves-on-étale-site-and-sheafification?noredirect=1#comment9171600_4386913

Answer 1

It's not diagonal. A section of $\mathcal F$ on $U$ has a germ at each element of $\text{Hom}_{X}(\bar{x},U)$, and you take the product of all those germs, which may be different.

It's not so hard to see that if you tried to do it diagonally, it wouldn't be functorial for automorphisms of $U$, say in the case when $U$ is a finite étale cover.