# Functorial representation of reduced schemes

Let $\mathsf{CRing}_{\mathsf{red}}$ denote the category of reduced commutative rings, and $\mathsf{Sch}_{\mathsf{red}}$ the category of reduced schemes. Let $L : [\mathsf{CRing}_{\mathsf{red}},\mathsf{Set}] \to [\mathsf{CRing},\mathsf{Set}]$ be the left Kan extension (for a sufficiently large version of $\mathsf{Set}$ on the right so that it exists), given by $L(X)(A) = \mathrm{colim}_{R \to A,\, R \text{ reduced}} \, X(R)$. We have a diagram of fully faithful functors:

$$\begin{array}{c} \mathsf{Sch}_{\mathsf{red}} & \rightarrow & \mathsf{Sch} \\ \downarrow && \downarrow \\ [\mathsf{CRing}_{\mathsf{red}},\mathsf{Set}] & \rightarrow & [\mathsf{CRing},\mathsf{Set}]. \end{array}$$

Does it commute? In other words, if $X$ is a reduced scheme and $A$ is an arbitrary commutative ring $A$, is the canonical map $$\mathrm{colim}_{R \to A,\, R \text{ reduced}} \, X(R) \longrightarrow X(A)$$ bijective? This is clear when $X$ is affine. It also holds when $A$ is local.