Let $S$ be a connected scheme. Let $FEt_S$ be the Galois category of schemes $X$ finite etale over $S$. Let $I$ be a directed set, and $\{C_i\}_{i\in I}$ a projective system of objects in $FEt_S$. I'm interested in considering morphisms from $\{C_i\}_{i\in I}$ to a fixed scheme $X$ (especially in the case where $X\in FEt_S$).
By definition, such morphisms are given by \begin{equation}Hom_{Pro-FEt_S}(\{C_i\}_{i\in I},X) = \varinjlim_{i\in I} Hom_{FEt_S}(C_i,X)\end{equation}
Taking limits gives a functor from $Pro-FEt_S$ to $Sch/S$, so we may also consider the limit $C := \varprojlim_{i\in I}C_i$ in $Sch/S$.
If $S$ is Noetherian and $X\in FEt_S$, then by http://stacks.math.columbia.edu/tag/01ZC , we have that $$Hom_{Sch/S}(C,X) = \varinjlim_{i\in I} Hom_{Sch/S}(C_i,X)$$ so taking limits identifies $Pro-FEt_S$ with the full subcategory of $Sch/S$ consisting of inverse limits of objects in $FEt_S$.
My questions are:
Is everything above correct?
Unfortunately I have almost zero intuition for non-Noetherian rings, so if $S$ is not Noetherian (but $X$ is still in $FEt_S$), can someone explain what can go wrong in this case? Can someone provide an example where $Hom_{Sch/S}(C,X) \ne \varinjlim_{i\in I} Hom_{Sch/S}(C_i,X)$ ?