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Why is lack of functoriality of the Lisse-Etale topology specific to the Lisse-Etale topology?

You are right, $f^{-1} \mathcal O_X \neq \mathcal O_Y$. in Zariski topology. Your Yoneda-type calculation doesn't apply in Zariski topology because sheaf $\mathcal O_X$ is not representable in Zariski ...
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