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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).

Proof. The equivalence of $(1)$ and $(2)$ follows by combining Example 7.5.2.9 with Corollary 7.5.5.15. The equivalence with $(3)$ follows by combining the same results with Proposition 6.3.1.13 and Example 7.5.2.6. $\square$