Corollary 9.2.9.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pushouts and let $(S_{L}, S_{R} )$ be a weak factorization system on $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The pair $(S_{L}, S_{R} )$ is a factorization system on $\operatorname{\mathcal{C}}$.
- $(2)$
For every $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z } \]
of $\operatorname{\mathcal{C}}$, if $f$ and $h$ belong to $S_{L}$, then $g$ also belongs to $S_{L}$.
- $(3)$
For every morphism $f: X \rightarrow Y$ which belongs to $S_{L}$, the relative codiagonal $\gamma _{X/Y}: Y \coprod _{X} Y \rightarrow Y$ also belongs to $S_{L}$.
Proof.
We first show that $(1) \Rightarrow (2)$. Assume that $(S_{L}, S_{R} )$ is a factorization system and consider a $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z } \]
of $\operatorname{\mathcal{C}}$. If $f$ and $h$ belong to $S_{L}$, then they are left orthogonal to $S_{R}$. Applying Corollary 9.2.7.15, we deduce that $g$ is also left orthogonal to $S_{R}$, so that $g \in S_{L}$ by virtue of Proposition 9.2.9.11.
We now show that $(2)$ implies $(3)$. Let $f: X \rightarrow Y$ be a morphism which belongs to $S_{L}$. Then the relative codiagonal $\gamma _{X/Y}$ fits into a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \coprod _{X} Y \ar [dr]^{ \gamma _{X/Y} } & \\ Y \ar [rr]^{ \operatorname{id}_{Y} } \ar [ur]^{f'} & & Y, } \]
where $f'$ is a pushout of $f$. Since $S_{L}$ is weakly saturated (Corollary 9.2.6.6), it contains the morphisms $f'$ and $\operatorname{id}_{Y}$. If condition $(2)$ is satisfied, then $\gamma _{X/Y}$ also contains $\gamma _{X/Y}$.
We now complete the proof by showing that $(3)$ implies $(1)$. Let $g$ be a morphism of $\operatorname{\mathcal{C}}$ which belongs to $S_{R}$. Then $g$ is weakly right orthogonal to $S_{L}$, and we wish to show that $g$ is right orthogonal to $S_{L}$. This follows by combining assumption $(3)$ with Corollary 9.2.7.21.
$\square$