Remark 5.3.2.8 (Functoriality). Let $\operatorname{\mathcal{C}}$ be a category. Then the formation of homotopy colimits determines a functor
\[ \underset { \longrightarrow }{\mathrm{holim}}: \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow (\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \quad \quad \mathscr {F} \mapsto \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ). \]
Moreover, this functor preserves small limits and colimits.