Corollary 4.4.3.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$ and $Y$. The following conditions are equivalent:
- $(1)$
The objects $X$ and $Y$ are isomorphic.
- $(2)$
There exists a connected Kan complex $\operatorname{\mathcal{E}}$, a pair of vertices $\overline{X}, \overline{Y} \in \operatorname{\mathcal{E}}$, and a morphism $f: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ satisfying $f( \overline{X}) = X$ and $f( \overline{Y} ) = Y$.
- $(3)$
There exists a contractible Kan complex $\operatorname{\mathcal{E}}$, a pair of vertices $\overline{X}, \overline{Y} \in \operatorname{\mathcal{E}}$, and a morphism $f: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ satisfying $f( \overline{X}) = X$ and $f( \overline{Y} ) = Y$.