$p : E \to X$ be a covering map. Then for some $x \in X$, the fiber $p^{-1}(x)$, by definition, is the pullback of the diagram $$\require{AMScd}
\begin{CD}
x \times_X E @>>> E\\
@VVV @VVV \\
x @>{i}>> X
\end{CD}$$ Where $i$ is the inclusion map of the point. I tried drawing the corresponding diagram for fields, and it looked like this $$\require{AMScd}
\begin{CD}
K \otimes_k \overline{k} @<<< K\\
@AAA @AAA \\
\overline{k} @<{i}<< k
\end{CD}$$ where $i$ is again the inclusion (the algebraic version of a "point", by the discussion we have had previously) and the leftmost map is the inclusion…