Most sources about motivic homotopy theory mention that the category of (smooth) separated schemes of finite type over a (Noetherian of finite Krull dimension) base $S$ is essentially small, which is essential for the homotopy theory of its simplicial sheaves. Yet, none of the sources I came across proves or references that this category is not small, or indicates the small category that it is equivalent to. I guess the latter should be the category of (smooth) finitely generated algebras over $\Gamma(S,\mathcal{O}\_S)$. 


I would like to ask for a reference where such a statement is proven, preferably in more generality, if it exists.



Smallness of the category of schemes of finite type
You can prove essential smallness of the category of finite type $S$-schemes (without further assumptions), where $S$ is any scheme, as follows:


* If $S$ is affine and we only consider affine finite type $S$-schemes, these correspond to finite type $\Gamma(S)$-algebras. These are isomorphic to algebras of the form $\Gamma(S)[x\_1,\dotsc,x\_n]/I$ for some natural number $n$ and some ideal $I$. These algebras form a set.
* If $S$ is affine and we allow separated finite type $S$-schemes, these may be glued from affine finite type $S$-schemes $X\_1,\dotsc,X\_n$, $n \in \mathbb{N}$, and isomorphisms between the affine schemes $X\_i \cap X\_j \subseteq X\_i$ and $X\_j \cap X\_i \subseteq X\_j$. By the affine case, we can classify them by a set.
* If $S$ is affine and we allow arbitrary finite type $S$-schemes, these may be glued from finitely many affine finite type $S$-schemes $X\_1,\dotsc,X\_n$, $n \in \mathbb{N}$, and isomorphisms between the separated schemes $X\_i \cap X\_j \subseteq X\_i$ and $X\_j \cap X\_i \subseteq X\_j$. By the first two cases, we can classify them by a set.
* If $S$ is arbitrary, then finite type $S$-schemes may be glued from finite type $U$-schemes $X\_U$, where $U$ runs through all open affines $U \subseteq S$, together with isomorphisms $X\_U \times\_U V \cong X\_V$ for $V \subseteq U$. By the affine case, we may classify them by a set.


Remark that the category of locally finite type $S$-schemes is only essentially small when $S=\emptyset$: Consider $\coprod\_{i \in I} S$ for sets $I$.


As Qiaochu points out, the category of (finite type) $S$-schemes is never small because it contains, for every set $X$, the scheme $(\emptyset,\mathcal{O})$ with the structure sheaf $\mathcal{O}(\emptyset)=(\{X\},+,\cdot)$.


