Kerodon

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Proposition 7.1.1.17. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $K$ be a simplicial set. Then:

• The $\infty$-category $\operatorname{\mathcal{C}}$ admits $K$-indexed limits if and only if the diagonal functor $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ admits a right adjoint $G$. If this condition is satisfied, then the right adjoint $G: \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ carries each diagram $u: K \rightarrow \operatorname{\mathcal{C}}$ to a limit $\varprojlim (u) \in \operatorname{\mathcal{C}}$.

• The $\infty$-category $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits if and only if the diagonal functor $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ admits a left adjoint $F$. In this condition is satisfied, then the left adjoint $F: \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ carries each diagram $u: K \rightarrow \operatorname{\mathcal{C}}$ to a colimit $\varinjlim (u) \in \operatorname{\mathcal{C}}$.

Proof. Apply Proposition 6.2.4.1. $\square$