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Proposition 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:


The diagram $\overline{\mathscr {F}}$ is a categorical colimit diagram.


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


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

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