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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?

This question was inspired by thinking about this fundamental question of Jonathan Sterling.

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.

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The answer to your question is unfortunately no. The terminal topos is not even finitely presentable.

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