Remark 1.1.4.12. The fully faithful embedding

\[ \operatorname{Set}\hookrightarrow \operatorname{Set_{\Delta }}\quad \quad S \mapsto \underline{S}_{\bullet } \]

preserves (small) limits and colimits (since limits and colimits of simplicial sets are computed levelwise; see Remark 1.1.1.13). It follows that the collection of discrete simplicial sets is closed under the formation of (small) limits and colimits in $\operatorname{Set_{\Delta }}$.