The setting for this question is the (2,1) category Topoi.


* 0-cells in Topoi are Grothendieck topoi.
* 1-cells are geometric morphisms and have the direction of the right adjoint.
* 2-cells are invertible natural transformation (between the left adjoints).


Johnstone (and Lurie) proves that this 2-category has pseudo colimits that can be computed in Cat. To be precise, given a diagram in Topoi its colimit is the limit of the corresponding diagram in Cat associated to the inverse images.



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> **Q.** Does the representable functor $\mathsf{pt}:= \text{Topoi(Set, )} $ preserve directed colimits?
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After some naive optimism, I have the feeling that the answer is *no*, but I suspect that it might be known.


* A candidate counterexample would be a topos $\mathcal{E}$ whose subtopoi lattice is not infinitary distributive, in fact a point is an atom in this coHeyting algebra.
* Another idea is to build a topos with with some points as a directed colimit of topoi without points, but I am not sure that this is possible at all.
* If a counterexample exist, I expect that it's possible to exhibit a *localic* counterexample.



Is Set a finitely presentable object in Topoi?
If $\mathcal{C}$ is a small category with finite limits then geometric morphisms from ${\rm Set}$ to the presheaf topos ${\rm PSh}(\mathcal{C})$ are in bijection with left exact functors $\mathcal{C} \to {\rm Set}$, or, equivalently, with objects in the pro-category ${\rm Pro}({\cal C})$. More explicitly, if ${\bf X} = \{X\_i\}\_{i \in {\cal I}}$ is a pro-object in ${\cal C}$ then the corresponding point ${\bf X}^\*:{\rm PSh}(\mathcal{C}) \to {\rm Set}$ sends a presheaf $F :\mathcal{C}^{\rm op} \to {\rm Set}$ to ${\rm colim}\_{i \in {\cal I}} F(X\_i)$. If ${\cal C}\_1,{\cal C}\_2$ are two categories with finite limits and $f: {\cal C}\_1 \to {\cal C}\_2$ is a functor (which does not necessarily preserve finite limits) then we have a geometric morphism $f\_\*: {\rm PSh}(\mathcal{C}\_1) \leftrightarrows {\rm PSh}(\mathcal{C}\_2): f^\*$ given by restriction and right Kan extension. We can then check that the functor on points ${\rm Pro}({\cal C}\_1) \to {\rm Pro}({\cal C}\_2)$ induced by $f\_\*$ sends $\{X\_i\}\_{i \in {\cal I}}$ to $\{f(X\_i)\}\_{i \in {\cal I}}$. 


If we now take a sequence
$$ {\cal C\_1} \to {\cal C\_2} \to ... \to {\cal C\_n} \to ... \to $$ 
of categories with finite limits (where the functors ${\cal C\_i} \to {\cal C\_{i+1}}$ are not assumed to preserve finite limits) then 
$$ {\rm colim}^{\rm Topoi}\_i {\rm PSh}({\cal C}\_i) \simeq {\rm lim}^{\rm Cat}\_i {\rm PSh}({\cal C}\_i) \simeq {\rm PSh}({\rm colim}\_i{\cal C\_i}), $$ 
but in general the map ${\rm colim}\_i{\rm Pro}({\cal C\_i}) \to {\rm Pro}({\rm colim}\_i{\cal C\_i})$ is not an equivalence. For example, it is often not essentially surjective: take ${\cal C\_n} = [n]$ with each consecutive map $[n] \to [n+1]$ being the inclusion as $[n] \cong \{1,...,n+1\} \subseteq [n+1]$ (these categories have finite limits because they are posets in which every finite subset has a minimal element).


