Let $Topos$ be the $(2,1)$-category of Grothendieck toposes and geometric morphisms. This is a $V$-sized, locally $V$-sized, locally locally small $(2,1)$-category with all small (2,1)-colimits (=pseudocolimits) and terminal object given by the topos $Set$.


**Question 1:** Is $Set$ a [tiny](https://ncatlab.org/nlab/show/tiny+object) object of $Topos$?


That is, does the covariant hom-functor $Pts = Topos(Set,-): Topos \to GPD$ preserve all small (2,1)-colimits?


**Question 2:** If so, what are some other tiny objects in $Topos$? If not, are there any at all?


This question was inspired by thinking about [this fundamental question of Jonathan Sterling](https://mathoverflow.net/questions/383887/describing-the-points-of-a-glued-topos).


I'd also be interested in the $\infty$-categorical versions of these questions.


I'm pretty sure that $Topos(Set,-)$ commutes with small coproducts. So it would suffice to consider pseudocoequalizers or pseudopushouts.


As a [De Morgan topos](https://ncatlab.org/nlab/show/De+Morgan+topos), Theorem D4.6.15 of the Elephant shows that $Set$ is projective with respect to proper separated localic surjections. But I'm not sure how those compare to descent morphisms, nor how close projectiveness with respect to descent morphisms would come to $Topos(Set,-)$ preserving descent objects.



Is $Set$ a tiny topos?
The answer to your question is unfortunately **[no](https://mathoverflow.net/questions/322396/is-set-a-finitely-presentable-object-in-topoi)**. The terminal topos is not even finitely presentable.


