ScalingStacks

Remark 7.1. In any $\infty$-category $\mathcal{C}$ which satisfies Axiom C.2 the cells detect equivalences. That is $f:X\to Y$ is an equivalence in $\mathcal{C}$ if and only if it induces equivalences $\map(C_{k},X)\to\map(C_{k},Y)$ for all $0\leq k\leq n$. This is clear, since for such a map the full subcategory of those $H\in\mathcal{C}$ such that $\map(H,X)\to\map(H,Y)$ is an equivalence is stable under colimits and contains the cells, and is thus all of $\mathcal{C}$.