Proposition 6.2.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, where $Y \in \operatorname{\mathcal{C}}'$. The following conditions are equivalent:
- $(1)$
The morphism $f$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$, in the sense of Definition 6.2.2.1.
- $(2)$
For every object $Z \in \operatorname{\mathcal{C}}'$, the restriction map $\operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \rightarrow \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Z\} $ is a homotopy equivalence of Kan complexes.
- $(3)$
The restriction map $u: \operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}'$ is an equivalence of $\infty $-categories.
- $(4)$
The restriction map $u$ is a trivial Kan fibration.
- $(5)$
For $n \geq 2$, every morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{C}}$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$, provided that $\sigma _0$ carries the initial edge $\Delta ^1 = \operatorname{N}_{\bullet }( \{ 0 < 1 \} )$ to the morphism $f$ and satisfies $\sigma _0(i) \in \operatorname{\mathcal{C}}'$ for $i \geq 2$.