Remark 7.7.2.18 (05TW)

Remark 7.7.2.18. Let $K$ be a simplicial set, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and $K$-indexed colimits, and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram having colimit $\varinjlim (F)$. Stated more informally, Theorem 7.7.2.8 asserts that the colimit of $F$ is universal if and only if the formation of pullbacks induces a fully faithful functor

\[ \operatorname{\mathcal{C}}_{ / \varinjlim (F) } \rightarrow \varprojlim _{x \in K} \operatorname{\mathcal{C}}_{ / F(x) } \quad \quad E \mapsto \{ F(x) \times _{ \varinjlim (F) } E \} _{x \in K} \]

(see Remark 7.7.2.7). Similarly, the colimit of $F$ is strongly universal if and only if this functor is an equivalence of $\infty $-categories. More precisely, if $\kappa $ is an uncountable cardinal for which $\operatorname{\mathcal{C}}$ is essentially small, then the colimit of $F$ is strongly universal if and only if it is preserved by the functor

\[ \operatorname{\mathcal{C}}\rightarrow (\operatorname{\mathcal{QC}}^{< \kappa })^{\operatorname{op}} \quad \quad C \mapsto \operatorname{\mathcal{C}}_{/C} \]

given by contravariant transport for the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$.