Inducing a coreflector $\mathsf{Sch} \to S$ from a reflector $\mathsf{CRing} \to C$

The following is a rephrasing of Hartshorne Chapter II Exercise 2.3:

For any ring $A$, let $A_{\text{red}}$ be the quotient of $A$ by its ideal of nilpotents. If $\mathcal{S}$ is a sheaf of rings on a topological space, let $\mathcal{S}_\text{red}$ be the sheafification of the presheaf $U \mapsto \mathcal{S}(U)_{\text{red}}$. We say a scheme is reduced if all of its section rings are reduced.

Let $X$ be a scheme with structure sheaf $\mathcal{O}$. Show that the ringed space $(X, \mathcal{O}_{\text{red}})$ is a reduced scheme, which we denote by $X_{\text{red}}$. Then show that for any reduced scheme $X$ and any scheme $Y$, there is a natural bijection $$ \operatorname{Hom}(X, Y) \cong \operatorname{Hom}(X, Y_{\text{red}}). $$

I solved this a while ago in a concrete and rather manual way, but now I'm suspecting that I have missed some slick formal argument. After all, $A \mapsto A_{\text{red}}$ and $X \mapsto X_{\text{red}}$ are just left and right adjoints of inclusions, respectively. We're reducing a scheme simply by reducing its section rings (and sheafifying). I would like to know which properties of the subcategory $\mathsf{CRingRed} \subset \mathsf{CRing}$ of reduced rings make this work quite naturally.

Here is one attempt to generalize the situation:

• Let $C$ be a full subcategory of $\mathsf{CRing}$. Suppose the inclusion $I : C \to \mathsf{CRing}$ has a left adjoint $F : \mathsf{CRing} \to C$.
• View (pre)sheaves of rings on a topological space as contravariant functors. If $\mathcal{S}$ is such a presheaf, denote its sheafification by $\mathcal{S}^+$.
• Let $\widetilde{G} : \mathsf{Sch} \to \mathsf{RS}$ be the (covariant) functor sending a scheme $X$ with structure sheaf $\mathcal{O}$ to the ringed space $(X, (I \circ F \circ \mathcal{O})^+)$. Here $\mathsf{RS}$ is the category of ringed spaces.

Question 1. When does $\widetilde{G}$ factor through $\mathsf{Sch}$? More importantly, when does it factor through a (full) subcategory $S \subset \mathsf{Sch}$, such that $G : \mathsf{Sch} \to S$ is a right adjoint of the inclusion $S \to \mathsf{Sch}$?

Question 2. In case $\widetilde{G}$ does factor through such $S$, is there a natural way to describe $S$? In the case $C = \mathsf{CRingRed}$, it turns out that $S$ is the full subcategory of schemes whose section rings (or whose stalks) lie on $C$. Is this always the case?

And an extra question. In this setting, does it serve us better to view schemes not as locally ringed spaces, but as sheaves on the Zariski site $\mathsf{CRing}^{\text{op}}$? I have learned this viewpoint only recently, so I can't tell what $G$ or $S$ should look like here, but maybe they turn out to be simpler...?

Something I don't like about the above setup is that to describe $\widetilde{G}$, I have to decouple the space and structure sheaf, do something to the sheaf, and then couple them back. I would much rather do something to the scheme directly. Perhaps there is a better way to do this.