Is the étale site a small category? Consider the étale site $X_{ét}$ of a scheme $X$. As a category, this is the collection of all étale schemes over $X$. Now, is this a set (i.e., is the étale site a small category)? If $X=Spec\ k$, one could suspect that every set has a scheme structure which is étale over $X$. Namely, $\coprod Spec \ k$, where the coproduct is taken over the cardinality of the set. If so, the class of étale schemes over $X$ would be a proper class. Is this true? Thank you in advance. P.S.: My question is motivated by the following: if $X_{ét}$ is a proper class, I'm afraid $Sets^{X_{ét}^{op}}$ shouldn't be a category. But the theory of étale cohomology and of the étale topos is based on the fact that its "subcategory" of sheaves on the étale site $Sh(X_{ét})$ is indeed a category (which we call the étale topos). So, either $Sh(X_{ét})$ while $Sets^{X_{ét}^{op}}$ is not, which would be strange to me, or we have a problem.

You are right: $X_{ét}$ is large (in fact, essentially large). This means that there is no category of presheaves on $X_{ét}$ in ZFC, or if you are using universes you would need to go to a higher universe to talk about the category of presheaves.

However, the category of _sheaves_ $Sh(X_{ét})$ is indeed a genuine category that can be defined without enlarging your universe. This is because there is a small set of objects of $X_{ét}$ which can be used to cover all other objects, and so a sheaf is uniquely determined by the values it takes on a small subcategory of $X_{ét}$. Indeed, since an étale map is locally of finite presentation, it suffices to consider affine schemes which are finitely presented étale covers of affine open subsets of $X$, and there is an (essentially) small set of these.