Remark 6.2.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a class of short morphisms for $\operatorname{\mathcal{C}}$. Let $f: X \rightarrow Y$ and $g: X \rightarrow Y$ be morphisms of $\operatorname{\mathcal{C}}$ which are homotopic. If $f$ belongs to $S$, then $g$ also belongs to $S$. This follows by applying property $(2)$ of Definition 6.2.5.4 to a $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{\operatorname{id}} & \\ X \ar [ur]^{g} \ar [rr]^{f} & & Y. } \]