$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark (Colimits). Let $\kappa $ be an infinite cardinal and let $\{ S_{i} \} _{i \in \operatorname{\mathcal{I}}}$ be a diagram of simplicial sets indexed by a category $\operatorname{\mathcal{I}}$. Suppose that the set of objects $\mathrm{Ob}( \operatorname{\mathcal{I}})$ has cardinality smaller than the cofinality of $\kappa $. Then the colimit $\varinjlim _{i \in \operatorname{\mathcal{I}}} S_ i$ is also $\kappa $-small (since it can be realized as a quotient of the coproduct $\coprod S_ i$, which is $\kappa $-small by virtue of Remark