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 $F$. If 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 $\varinjlim (u) \in \operatorname{\mathcal{C}}$.