Example 4.7.3.13 (Successor Cardinals). Let $\kappa $ be an infinite cardinal and let $\kappa ^{+}$ be its successor (Example 4.7.2.11). Then a set $S$ is $\kappa ^{+}$-small if and only if it has cardinality $\leq \kappa $. It follows that $\kappa ^{+}$ is a regular cardinal. That is, if $\{ T_ s \} _{s \in S}$ is a collection of sets of cardinality $\leq \kappa $ indexed by a set $S$ of cardinality $\leq \kappa $, then the disjoint union ${\coprod }_{s \in S} T_{s}$ also has cardinality $\leq \kappa $. To prove this, choose a collection of monomorphisms $\{ i_ s: T_ s \hookrightarrow T \} _{s \in S}$, where $T$ is a set of cardinality $\kappa $. We then obtain a monomorphism
where the set $S \times T$ has cardinality $\leq \kappa $ by virtue of Proposition 4.7.3.5.