Variant 6.2.2.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between (small) $\infty $-categories, which we regard as a morphism in the $\infty $-category $\operatorname{\mathcal{QC}}$. For every integer $n \geq -1$, the following conditions are equivalent:
- $(1_ n)$
The functor $F$ exhibits $\operatorname{\mathcal{D}}$ as a local $(n-1)$-truncation of $\operatorname{\mathcal{C}}$, in the sense of Definition 4.8.2.9.
- $(2_ n)$
The functor $F$ exhibits $\operatorname{\mathcal{D}}$ as a $\operatorname{\mathcal{QC}}_{\leq n}$-reflection of $\operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{QC}}_{\leq n} \subset \operatorname{\mathcal{QC}}$ denotes the full subcategory spanned by those $\infty $-categories which are locally $(n-1)$-truncated.
This follows from characterization of local $(n-1)$-truncations supplied by Proposition 4.8.2.18 (together with Remark 5.5.4.5). For example, if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then we can regard the homotopy $n$-category $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ as a $\operatorname{\mathcal{QC}}_{\leq n}$-reflection of $\operatorname{\mathcal{C}}$ (Corollary 4.8.4.8). It follows that $\operatorname{\mathcal{QC}}_{\leq n}$ is a reflective subcategory of $\operatorname{\mathcal{QC}}$.