Kerodon

Remark 7.1.1.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let $Y \in \operatorname{\mathcal{C}}$ be an object. If $\alpha : \underline{Y} \rightarrow u$ is a natural transformation, then the condition that $\alpha $ exhibits $Y$ as a limit of $u$ depends only on its homotopy class $[\alpha ]$ (as a morphism in the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$). Similarly, if $\beta : u \rightarrow \underline{Y}$ is a natural transformation, then the condition that $\beta $ exhibits $Y$ as a colimit of $u$ depends only on its homotopy class $[\beta ]$.