Suppose I have a scheme $X$ and I want to know if $X$ is reduced, but all I have access to is the functor $$ R\mapsto X(R)=Mor(\operatorname{Spec}(R),X) $$
from commutative rings to sets (rather than, say, an explicit covering by spectra of some rings). Is there some nice criterion for this?
Some thoughts: The universal property of the reduction is that every map from a reduced scheme $Y$ to $X$, factors uniquely through $X_{red}$. Thus, reducedness of $X$ seems to be "in the opposite direction" in the sense that it says something about morphisms from $X$ (that they always factor through the reduction of the target) rather than morphisms to $X$.