Kerodon

Remark 6.2.5.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $S$ be a class of short morphisms of $\operatorname{\mathcal{C}}$, and let $f: X \rightarrow Z$ be a morphism of $\operatorname{\mathcal{C}}$, which we identify with an object $\widetilde{X}$ of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Z}$. Then an $S$-optimal factorization of $f$ can be viewed as a morphism $\widetilde{X} \rightarrow \widetilde{Y}$ in $\operatorname{\mathcal{C}}_{/Z}$ which exhibits $\widetilde{Y}$ as a $\operatorname{\mathcal{C}}^{\mathrm{short}}_{/Z}$-reflection of $\widetilde{X}$, in the sense of Definition 6.2.2.1. Consequently, condition $(3)$ of Definition 6.2.5.4 is equivalent to the requirement that the full subcategory $\operatorname{\mathcal{C}}^{\mathrm{short}}_{/Z} \subseteq \operatorname{\mathcal{C}}_{/Z}$ is reflective, for each object $Z \in \operatorname{\mathcal{C}}$.