Suppose that $\mathsf{C}$ is a cocomplete category and $F : \mathsf{C} \to \mathsf{C}$ is and endofunctor together with a natural transformation $\eta : \mathsf{C} \to 1$. Then, for any $x \in X$ one can form the diagram
$$ x \xleftarrow{\eta} Fx \xleftarrow{F \eta} F^2x \xleftarrow{F^2\eta} F^3x \leftarrow \cdots \tag{$\star$} $$
which is an object in $\mathsf{pro-C}$. Now, given an arrow $f : x \to y$ in $\mathsf{C}$ then we have a natural transformation between the constructions in $(\star)$ via $F^kf : F^kx \to F^ky$.
Similarly we have a construction for $\mathsf{ind-C}$ when having a natural transformation $\xi : 1 \to \mathsf{C}$. If I am not mistaken, the natural transformations constitute morphisms between the pro/ind objects.
For example, in general given $X,Y : I \to \mathsf{C}$ two ind-objects with common domain, then a natural transformation $\mu : X \Rightarrow Y$ gives maps $\mu_i : X_i \to Y_i$ and so their equivalence classes in $\mathsf{colim}_J(X_i,Y_j)$ should assemble into an element of $\mathsf{lim} \ \mathsf{colim}\hom(X_i,Y_j) = \hom(X,Y)$.
In particular, for the first case, we have a functor
$$ \widetilde{F} : \mathsf{C} \to \mathsf{pro-C}. $$
If $F$ preserves (co)limits, does $\widetilde{F}$ preserve (co)limits? What about the case for ind-objects?
The motivation for this question comes from the baricentric subdivision functor on simplicial sets. Since it is a left adjoint, it preserves all colimits, and I am interested in knowing if the extension to pro-simplicial sets does too.
Edit: for the purposes of what I am trying to understand, the following more concrete question suffices.
Given $K$ a simplicial set, we can for its baricentric subdivision
$$ \mathsf{sd} K = \lim_{\Delta^n \to K} \mathsf{sd} \Delta^n $$
where $\mathsf{sd} \Delta^n$ is the simplicial set associated with the baricentric subdivision of $\Delta^n$ as a simplicial complex.
The so called 'last vertex map' gives a natural transformation $\ell : \mathsf{sd} \to 1$, hence for any simplicial set $K$ we have a pro-simplicial set
$$ \mathsf{sd}^\bullet K := K \xleftarrow{\ell} \mathsf{sd} K \xleftarrow{\mathsf{sd}\ell} \mathsf{sd}^2 K \leftarrow \cdots $$
and as described, a functor
$$ \mathsf{sd}^{\bullet}(-) : \mathsf{sSet} \to \mathsf{pro-sSet}. $$
Does $\mathsf{sd}^\bullet$ send finite colimits to finite colimits? Can changing the codomain to $\mathsf{sSet}^{\mathbb{N}^{op}}$ make a difference?