Proposition 7.5.8.4. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. The following conditions are equivalent:
- $(1)$
The diagram $\overline{\mathscr {F}}$ is a categorical colimit diagram.
- $(2)$
For every $\infty $-category $\operatorname{\mathcal{D}}$, the diagram of $\infty $-categories
\[ ( \operatorname{\mathcal{C}}^{\triangleright } )^{\operatorname{op}} \rightarrow \operatorname{QCat}\quad \quad C \mapsto \operatorname{Fun}( \overline{\mathscr {F}}(C), \operatorname{\mathcal{D}}) \]is a categorical limit diagram (Definition 7.5.5.1).
- $(3)$
For every $\infty $-category $\operatorname{\mathcal{D}}$, the diagram of Kan complexes
\[ ( \operatorname{\mathcal{C}}^{\triangleright } )^{\operatorname{op}} \rightarrow \operatorname{Kan}\quad \quad C \mapsto \operatorname{Fun}( \overline{\mathscr {F}}(C), \operatorname{\mathcal{D}})^{\simeq } \]is a homotopy limit diagram (Definition 7.5.4.1).