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

Theorem 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 Apply Proposition in the case $S = \Delta ^0$. $\square$