I will first state the general version of my question, but I do have a specific context in mind in which second I'll dance around.
(1.) Let $\mathsf{C}$ be a full subcategory of a category $\mathsf{D}$. Suppose $\{F_{m}\colon\mathsf{D}\to\mathsf{Set}\mid m\in\mathbf{N}\}$ is an inverse system of functors. Write $\lim_{m}F_{m}$ for the limit of $F_{m}$ taken in the category $\mathsf{D}$.
Suppose for every $m$, $F_{m}$ is determined by its values taken on inputs from $\mathsf{C}$. What conditions are necessary to ensure that $\lim_{m}F_{m}$ is determined by $\mathsf{C}$? Does this property have a name?
(2.) In my specific context, fix a commutative ring $R$ and let $\mathsf{D}=\mathsf{Alg}_{R}$ be the category of $R$-algebras. Let $\mathsf{C}=\mathsf{Noeth}_{R}$ be the full subcategory of noetherian $R$-algebras. It is known that if $Y$ is a locally noetherian scheme, that the functor of points $h_{Y}\colon\mathsf{Alg}_{R}\to\mathsf{Set}$, $h_{Y}(A):=\mathrm{Hom}_{\mathsf{Sch}_{R}}(\mathrm{Spec}A,Y)$, is determined by noetherian algebras $A$. (See Eisenbud-Harris' The Geometry of Schemes Exer. VI-3.) In general, if $\{X_{m}\mid m\in\mathbf{N}\}$ is an inverse system of $R$-schemes, then $\lim_{m}h_{X_{m}}$ is not necessarily representable, because the category $\mathsf{Sch}_{R}$ is not complete.
However, is it still the case that the functor $\lim_{m}h_{X_{m}}\colon\mathsf{Alg}_{R}\to\mathsf{Set}$ is determined by its values on noetherian $R$-algebras?