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.