# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark 7.1.1.9. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. Suppose we are given a natural transformation of diagrams $\beta : \underline{Y} \rightarrow u$, and let $\alpha : \underline{X} \rightarrow u$ be a composition of $\beta$ with the constant natural transformation $\underline{f}: \underline{X} \rightarrow \underline{Y}$. Then any two of the following three properties imply the third:

• The natural transformation $\alpha$ exhibits $X$ as a limit of the diagram $u$.

• The natural transformation $\beta$ exhibits $Y$ as a limit of the diagram $u$.

• The morphism $f: X \rightarrow Y$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{C}}$.