Kerodon

Remark 4.1.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a subcategory. Suppose we are given a pair of morphisms $f,g: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ having the same source and target. If $f$ and $g$ are homotopic as morphisms in the $\infty $-category $\operatorname{\mathcal{C}}$ and $f$ belongs to the subcategory $\operatorname{\mathcal{C}}'$, then $g$ also belongs to the subcategory $\operatorname{\mathcal{C}}'$ and the morphisms $f$ and $g$ are homotopic in the $\infty $-category $\operatorname{\mathcal{C}}'$. This a special case of Remark 4.1.2.7 (note that $f$ and $g$ are homotopic if and only if $g$ is a composition of $f$ with the identity morphism $\operatorname{id}_{Y}$; see Definition 1.4.3.1).