Is Category of schemes locally small?

Question

One can use Yoneda lemma in the context of locally small category. due to yoneda lemma, we can consider a scheme X as a functor of points. (like SpecA $\mapsto$ X(A)=Hom(SpecA, X) ). but I don't know that a scheme category is locally small. I guess the scheme category is not locally small. This curiocity comes out when I study group scheme theory. Yoneda lemma can be applied to only group scheme over a field k? Please feel free to write your opinion. Thank you.