Is there a way to phrase "there does not exist a universal set" in structural language? Both ZFC (which is a good example of a material set theory) and ETCS (which is a good example of a structural set theory) prove the sentence "there is no set having maximum cardinality" as an easy corollary of Cantor's theorem. However, ZFC also proves the sentence "there is no universal set," or more precisely: $$\neg \exists V \forall x(x \in V).$$
The question then arises of whether we can phrase this proposition for a structural set-theory like ETCS.
Question. Is there a way to express "there does not exist a universal set" in structural language?

There is no real difference between the two, from a structural perspective. More precisely:

Theorem. Let $\mathcal{S}$ be an elementary topos. If there is an object $V$ in $\mathcal{S}$ such that, for all objects $X$ in $\mathcal{S}$, there is a monomorphism $X \rightarrowtail V$, then $\mathcal{S}$ is the degenerate topos.

See here.
