A Grothendieck's universe is such a set $U$ so that

$\forall x \in U, x \subseteq U$,

$\forall x,y \in U, \{x,y\} \in U$,

$\forall x \in U, \mathcal{P}(x) \in U$,

given a family $(X_i)_{i \in I}$ such that $I \in U$ and any $X_i \in U$ we have $\bigcup_{i \in I} X_i$,

$\mathbb{N} \in U$.

We can "do set theory" inside a single Grothendieck's universe. However, the existence of a single universe wasn't enough for Grothendieck, he introduced the axiom of universes:

Let $X$ be a set. Then there is a universe $U$ such that $X \in U$.

In particular, this guarantees that given a universe $U$, there is always a universe $V$ so that $U \in V$, hence we can enlarge a given universe.

Now, I'm personally interested in not the universe itself but it's enhancement based on a model-theoretic concept of "elementary substructure". Here Michael Shulman introduces a theory which he calls $ZMC/S$ (which is based on S.Feferman's theory $ZFC/S$) which takes $ZFC$ and adds a constant symbol $S$ which is a set which is both a universe and an which satsfies the following reflection principle:

Let $\phi(x_1,...,x_n)$ be a formula in the language of set theory. Then

$$\forall x_1...\forall x_n, (\phi(x_1,...,x_n) \Longleftrightarrow \phi^S(x_1,...,x_n)),$$

where $\phi^S(x_1,...,x_n)$ denoted the formula obtained from $\phi(x_1,...,x_n)$ by restricting all quantifiers to the set $S$.

Assume this is the approach we wish to take for the foundations of category theory (I do not wish to discuss whether it is useful with respect to ordinary universes). Is a single such set $S$ enough for all purposes of category theory or do we instead need a proper class of similar universes, such as introduced there? Note that both Michael Shulman in his notes and Joel David Hamkins in his aforementioned answer claim that these theories are both equiconsistent with the assumption that "$Ord$ is Mahlo", which would imply that that they are equiconsistent with each other. However, consistency is not what interests me right now, what I want to understand is there there a matter of enlarging a universe when working with a single Feferman-Shulman universe $S$ as it is when working with a single Gorthendieck universe?