The category of schemes is not small-concrete.

Let S be a generating set. Let U be the set of all rings A such that Spec A is an open subset of a scheme in S. Let X be a set whose cardinality is larger than any element of U, for example, 2^{\bigsqcup_{A \in U} A}. Let K be the field Q(t_x)_{x \in X}, where t_x are a collection of algebraically independent generators indexed by X. So |K| is larger than |A| for any A in U. Since ring maps from a field are always injective, Hom(Spec A, Spec K)={} for every A in U, and therefore Hom(s, Spec K)={} for every s in S.

There is only one map from the empty set to itself. But Spec K has nontrivial isomorphisms, coming from permuting the generators. So

Hom(Spec K, Spec K) ---> SetHom( \bigsqcup_{s \in S} (Spec K)(S), \bigsqcup_{s \in S} (Spec K)(S))

is not injective.

The category of schemes is not small-concrete.

Let $S$ be a generating set. Let $U$ be the set of all rings $A \neq 0$ such that $\mathrm{Spec}(A)$ is an open subscheme of a scheme in $S$. Let $X$ be a set whose cardinality is larger than any element of $U$, for example, $2^{\bigsqcup_{A \in U} A}$. Let $K$ be the field $\mathbb{Q}(t_x)_{x \in X}$, where $t_x$ are a collection of algebraically independent generators indexed by $X$. So $|K|$ is larger than $|A|$ for any $A \in U$. Since ring maps from a field to a nontrivial ring are always injective, $\mathrm{Hom}(\mathrm{Spec}(A),\mathrm{Spec}(K))=\emptyset$ for every $A \in U$, and therefore $\mathrm{Hom}(s,\mathrm{Spec}(K))=\emptyset$ for every $s \in S$.

There is only one map from the empty set to itself. But $\mathrm{Spec}(K)$ has nontrivial isomorphisms, coming from permuting the generators. So

$\mathrm{Hom}(\mathrm{Spec}(K),\mathrm{Spec}(K)) \longrightarrow \mathrm{Hom}_{\mathrm{Set}^{S^\mathrm{op}}}( (\mathrm{Spec}(K))(-), (\mathrm{Spec}(K))(-))$

is not injective.

The category of schemes is not small-concrete.

Let S be a generating set. Let U be the set of all rings A such that Spec A is an open subset of a scheme in U. Let X be a set whose cardinality is larger than any element of U, for example, 2^{\bigsqcup_{A \in U} A}. Let K be the field Q(t_x)_{x \in X}, where t_x are a collection of algebraically independent generators indexed by X. So |K| is larger than |A| for any A in U. Since ring maps from a field are always injective, Hom(Spec A, Spec K)={} for every A in U, and therefore Hom(s, Spec K)={} for every s in S.

There is only one map from the empty set to itself. But Spec K has nontrivial isomorphisms, coming from permuting the generators. So

Hom(Spec K, Spec K) ---> SetHom( \bigsqcup_{s \in S} (Spec K)(S), \bigsqcup_{s \in S} (Spec K)(S))

is not injective.

The category of schemes is not small-concrete.

Let S be a generating set. Let U be the set of all rings A such that Spec A is an open subset of a scheme in S. Let X be a set whose cardinality is larger than any element of U, for example, 2^{\bigsqcup_{A \in U} A}. Let K be the field Q(t_x)_{x \in X}, where t_x are a collection of algebraically independent generators indexed by X. So |K| is larger than |A| for any A in U. Since ring maps from a field are always injective, Hom(Spec A, Spec K)={} for every A in U, and therefore Hom(s, Spec K)={} for every s in S.

There is only one map from the empty set to itself. But Spec K has nontrivial isomorphisms, coming from permuting the generators. So

Hom(Spec K, Spec K) ---> SetHom( \bigsqcup_{s \in S} (Spec K)(S), \bigsqcup_{s \in S} (Spec K)(S))

is not injective.