# Is $Set$ a tiny topos?

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 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?

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, 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.