Kerodon

Remark 7.1.1.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $Y$, let $K$ be a simplicial set, and let $\beta : u \rightarrow u'$ be an isomorphism in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Suppose we are given a natural transformation $\alpha : \underline{Y} \rightarrow u$, and let $\alpha ': \underline{Y} \rightarrow u'$ be any composition of $\alpha $ with $\beta $. Then $\alpha $ exhibits $Y$ as a limit of $u$ if and only if $\alpha '$ exhibits $Y$ as a limit of $u'$. Similarly, if $\gamma ': u' \rightarrow \underline{Y}$ is a natural transformation and $\gamma : u \rightarrow \underline{Y}$ is a composition of $\beta $ with $\gamma '$, then $\gamma $ exhibits $Y$ as a colimit of $u$ if and only if $\gamma '$ exhibits $Y$ as a colimit of $u'$.