# Kerodon

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Theorem 4.4.4.4. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $f,f': X \rightarrow \operatorname{\mathcal{C}}$ be diagrams in $\operatorname{\mathcal{C}}$ indexed by a simplicial set $X$, and let $u: f \rightarrow f'$ be a natural transformation. Then $u$ is a natural isomorphism if and only if, for every vertex $x \in X$, the induced map $\operatorname{ev}_{x}(u): f(x) \rightarrow f'(x)$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{C}}$.

Proof of Theorem 4.4.4.4. Apply Proposition 4.4.4.9 in the case $S = \Delta ^0$. $\square$