The Stacks project

Remark 59.56.4. Another way to state the conclusion of Theorem 59.56.3 and Fundamental Groups, Lemma 58.2.2 is to say that every sheaf on $\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}$ is representable by a scheme $X$ étale over $\mathop{\mathrm{Spec}}(K)$. This does not mean that every sheaf is representable in the sense of Sites, Definition 7.12.3. The reason is that in our construction of $\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}$ we chose a sufficiently large set of schemes étale over $\mathop{\mathrm{Spec}}(K)$, whereas sheaves on $\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}$ form a proper class.

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