Kerodon

Remark 7.1.1.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $Y$ and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then a natural transformation $\alpha : \underline{Y} \rightarrow u$ exhibits $Y$ as a limit of $u$ if and only if it exhibits $Y$ as a colimit of the induced diagram $u^{\operatorname{op}}: K^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$, when regarded as a morphism in the $\infty $-category $\operatorname{Fun}( K^{\operatorname{op}}, \operatorname{\mathcal{C}}^{\operatorname{op}} ) \simeq \operatorname{Fun}(K, \operatorname{\mathcal{C}})^{\operatorname{op}}$.