Let $\mathcal{F}:\mathrm{Sch}/S \to \{\mathrm{Sets}\}$ be a representable functor. Denote by $X$ the scheme representing $\mathcal{F}$. The question is whether the natural tranformation $\mathcal{F}(-) \to \mathrm{Hom}\_S(-,X)$ commute with inverse limits? More precisely, consider an inverse system of affine schemes $\{Z\_i\}\_{i \in I}$ with limit $Z:=\varprojlim\limits\_i Z\_i$. Consider a sequence of objects $(a\_i)\_{i \in I}$ with $a\_i \in \mathcal{F}(Z\_i)$ and $f\_i \in \mathrm{Hom}\_S(Z\_i,X)$ corresponding to each $a\_i$ (under the natural transformation). Suppose that the sequence $(f\_i)$ (resp. $(a\_i)$) converge to some $f \in \mathrm{Hom}\_S(\varprojlim\limits\_i Z\_i,X)$ (resp. $a \in \mathcal{F}(\varprojlim\limits\_i Z\_i)$). Is it true that $a$ must map to $f$ under the above natural transformation? 


I do not know if this question is obvious, in which case a reference will be sufficient.



Representable functors and direct limits
This question seems to have a few confusions in it. Here are three thoughts that might be helpful. (I'm guessing you are after a version of statement (2).)


(1) "[T]he natural transformation $\mathcal F(-) \to \mathrm{Hom}\_S(-,X)$" is an isomorphism. This is what it *means* when you say "$X$ represents $\mathcal F$". Isomorphism commute with *everything* categorical.


(2) Normally we think about *functors* commuting with limits, not natural transformations. If $\mathcal F$ is any contravariant functor and $\{X\_i\}\_{i\in I}$ is any diagram, then there is a canonical comparison map
$$ \varinjlim\_i \mathcal F(X\_i) \to \mathcal F(\varprojlim\_i X\_i). $$
$\mathcal F$ "takes limits to colimits" if this comparison map is an isomorphism for every diagram.


Suppose that $\mathcal F$ and $\mathcal G$ are two contravariant functors and that $\eta : \mathcal F \to \mathcal G$ is a natural transformation. Given a diagram $\{X\_i\}\_{i\in I}$, naturality of $\eta$ says that the following diagram commutes:
$$ \begin{matrix} \mathcal F(\varprojlim X\_i) & \overset{\eta(\varprojlim X\_i)}\longrightarrow & \mathcal G(\varprojlim X\_i) \\ \uparrow & & \uparrow \\ \varinjlim \mathcal F(X\_i) & \overset{\varinjlim \eta(X\_i)}\longrightarrow & \varinjlim \mathcal G(X\_i)\end{matrix} $$
If both $\mathcal F$ and $\mathcal G$ take limits to colimits, then the vertical arrows are isos. If $\eta$ is an isomorphism, then the horizontal arrows are isos. Regardless, this is perhaps a statement of the form "natural transformations commute with (co)limits."


(3) Representable functors always take colimits to limits, but only very rarely take limits to colimits. Doing so is analogous in representation theory to the representing object being an injective module.


