$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 6.2.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}'$ be a full subcategory, and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which satisfies the following conditions:
- $(0)$
Every morphism $u: X \rightarrow Y$ which belongs to $W$ is a $\operatorname{\mathcal{C}}'$-local equivalence.
- $(1)$
Every isomorphism of $\operatorname{\mathcal{C}}$ is contained in $W$.
- $(2_{-})$
Suppose we are given a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ & Y \ar [dr]^{v} & \\ X \ar [ur]^{u} \ar [rr]^{w} & & Z } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$, where $v$ belongs to $W$. Then $u$ belongs to $W$ if and only if $w$ belongs to $W$.
- $(3)$
For every object $X \in \operatorname{\mathcal{C}}$, there exists a morphism $u: X \rightarrow Y$ which belongs to $W$, where the object $Y$ is contained in $\operatorname{\mathcal{C}}'$.
Then $\operatorname{\mathcal{C}}'$ is a reflective subcategory of $\operatorname{\mathcal{E}}$ and $W$ is the collection of $\operatorname{\mathcal{C}}'$-local equivalences.
Proof.
It follows immediately from conditions $(0)$ and $(3)$ that the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is reflective. Let $u: X \rightarrow Y$ be a $\operatorname{\mathcal{C}}'$-local equivalence; we wish to show that $u$ is contained in $W$ (the converse follows from $(0)$). Applying condition $(0)$, we can choose morphisms $v_{X}: X \rightarrow X'$ and $v_{Y}: Y \rightarrow Y'$ which belong to $W$, where $X'$ and $Y'$ are contained in the subcategory $\operatorname{\mathcal{C}}'$. Then $v_{X}$ is a $\operatorname{\mathcal{C}}'$-local equivalence, so there exists a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ X \ar [r]^{u} \ar [d]^{v_ X} & Y \ar [d]^{v_{Y} } \\ X' \ar [r]^{u'} & Y' } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$. Since $u$ is a $\operatorname{\mathcal{C}}'$-local equivalence, it follows that $u'$ is also a $\operatorname{\mathcal{C}}'$-local equivalence, and therefore an isomorphism (Remark 6.2.2.4). It follows $(1)$ that the morphism $u'$ belongs to $W$. Applying condition $(2_{-})$ twice, we conclude that $u$ also belongs to $W$.
$\square$