Let $f\colon X\to Y$ be a morphism of schemes. Let $F_X$ and $F_Y$ be the contravariant functors from the category $Sch$ of schemes to the category of sets defined via the Yoneda construction, i.e. $F_X(S)=Hom_{Sch}(S,X)$. Let $\hat f\colon F_X\to F_Y$ be the corresponding morphism of functors induced by $f$.

Can one say in categorical terms (i.e. in terms of $\hat f$) when exactly the morphism $f$ is a closed imbedding?

All schemes may be assumed to be finitely presented over complex numbers, but possibly it is not very important.

I apologize if this question is too elementary or naive for MO.


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