I'm trying to prove that an étale map of spaces has discrete fibers. The first diagram I drew is:
$$\require{AMScd} \begin{CD} f^\ast\coprod_i \left\{ x \right\} @>>> f^\ast \coprod_i U_i @>>> Y\\ @VVV @VVV @VV{f}V\\ \left\{ x \right\} @>>> \coprod_iU_i @>>> X \end{CD}$$
However, I can't really do anything with it. However, if I pretend $f^\ast \coprod_iU_i\cong \coprod_if^\ast U_i$, I end up with the diagram below.
$$\require{AMScd} \begin{CD} \coprod_if_i^\ast \left\{ x \right\}@>>> \coprod_if^\ast U_i @>>> Y\\ @VVV @V{\coprod_if_i}VV @VV{f}V\\ \left\{ x \right\} @>>> \coprod_iU_i @>>> X \end{CD}$$
First, by using the pullback square on the right, I pass to the $f_i$'s which are homeomorphisms $f_i:f^\ast U_i\cong U_i$. Then using the left pullback I take the fibers of these $f_i$ at once. Since each $f_i$ is a homeo, $f_i^\ast \left\{ x \right\} \cong \left\{ x \right\}$, the top left corner is discrete, and is isomorphic to $f^\ast \left\{ x \right\}$ by pullback pasting.
I think my reasoning is justified in any extensive category, but I am not sure if so and exactly how.
Is my proof correct, and if so, how does $f^\ast \coprod_iU_i\cong \coprod_if^\ast U_i$ follow from extensivity?
Added. I just realized that my "definition" of étale was wrong. $f:Y\rightarrow X$ being étale means there's a cover of $Y$ on whose components $f$ is an iso, but this cover need not be the pullback of some other cover of $X$. Hence this is not the same as asking for a cover of $Y$ such that $f$ pulled back to its components is an iso...
Maybe the key is to realize the squares on the left are all pullbacks iff the right one is, which I think is true, but I don't see how it this recovers the fiber of $f$...
$$\require{AMScd} \begin{CD} f_i^\ast \left\{ x \right\}@>>> U_i \\ @VVV @VV{f_i}V\\ \left\{ x \right\} @>>> X \end{CD}\;\;\;\;\begin{CD} \coprod_i f_i^\ast \left\{ x \right\}@>>> \coprod_i U_i \\ @VVV @VV{\coprod_if_i}V\\ \left\{ x \right\} @>>> X \end{CD}$$
Maybe somehow using the factorization 
(source: presheaf.com)
for the horizontal map étale?
