Remark Let $f: X \rightarrow Y$ and $g: Y \rightarrow X$ be morphisms in an $\infty $-category $\operatorname{\mathcal{C}}$. Then $g$ is left homotopy inverse to $f$ if and only if $f$ is right homotopy inverse to $g$. Both of these conditions are equivalent to the existence of a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ satisfying $d_0(\sigma ) = g$, $d_1(\sigma ) = \operatorname{id}_{X}$, and $d_2(\sigma ) = f$, as depicted in the diagram

\[ \xymatrix { & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{\operatorname{id}_ X} & & X. } \]