The following is taken from Borceux Handbook of Categorical Algebra p.59:
Proposition 2.71. Consider the category $\mathcal{C}$ such that, for every category $\mathcal{D}$ and functor $F:\mathcal{D}\rightarrow\mathcal{C}$, the limit of $F$ exists. In that case, $\mathcal{C}$ is just a preordered set.
Proof. Let us use the axiom system of universes (see section 1.1) so that the objects of $\mathcal{C}$ constitute a set in some universe...
In section 1.1, we find the following axiom:
Axiom 1.1.4 Every set belongs to some universe.
Now how does this axiom imply that, for any arbitrary category $\mathcal{C}$, there exists a universe in which $\text{Ob}(\mathcal{C})$ is a set? (What does this even mean? Does this merely mean that for an arbitrary category $\mathcal{C}$ there exists a universe $\mathcal{U}$ such that $\text{Ob}(\mathcal{C})\in\mathcal{U}$?)