The common theme amongst these properties is that the aim to express closure of the set \(\mathcal U\) under various standard set-theoretic operations. This allows us to treat \(\mathcal U\) as a universe of sets in which we can carry out normal set-theoretic constructions without leaving the universe.
The Axiom of Universes is the general convention which is adopted in category theory and other fields which states that for every set \(x\) there exists a Grothendieck universe \(\mathcal U\) such that \(x \in \mathcal U\).
An element of \(\mathcal U\) is by convention called a \(\mathcal U\)-small set or simply a small set when the universe is clear from context or apparently also just a set (1).
A subset of \(\mathcal U\) is called \(\mathcal U\)-moderate or simply moderate when the universe is clear from context.
The general line of reasoning, at least in ZFC is that for every set \(x\) we assume there exists a Grothendieck universe \(\mathcal U\) such that \(x \in \mathcal U\). Accordinly for every universe \(\mathcal U\) there exists a larger universe \(\mathcal U'\) such that \(\mathcal U \in \mathcal U'\), and therefore also \(U \subseteq U'\). Thus under this assumption with have the guarantee of a hierarchy of universes (4):
\[
\mathcal U_0 \in \mathcal U_1 \in \mathcal U_2 \in \cdots
\]
Which are also cumulative in the sense that
\[
\mathcal U_0 \subseteq \mathcal U_1 \subseteq \mathcal U_2 \subseteq \cdots
\]