Proposition 1.5.4.5. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts, let $T$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $S$ be the collection of all morphisms of $\operatorname{\mathcal{C}}$ which are weakly left orthogonal to $T$. Then $S$ is closed under pushouts.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Suppose we are given a pushout diagram $\sigma :$
\[ \xymatrix@C =40pt@R=40pt{ A \ar [d]^{f} \ar [r]^-{s} & A' \ar [d]^{f'} \\ B \ar [r]^-{t} & B' } \]
where $f$ belongs to $S$. We wish to show that $f'$ also belongs to $S$. For this, we must show that every lifting problem
\[ \xymatrix@C =40pt@R=40pt{ A' \ar [d]^{f'} \ar [r]^-{u} & X \ar [d]^{g} \\ B' \ar [r]^-{v} \ar@ {-->}[ur] & Y } \]
admits a solution, provided that the morphism $g$ belongs to $T$. Using our assumption that $\sigma $ is a pushout square, we are reduced to solving the associated lifting problem
\[ \xymatrix@C =40pt@R=40pt{ A \ar [d]^{f} \ar [r]^-{u \circ s} & X \ar [d]^{g} \\ B \ar [r]^-{v \circ t} \ar@ {-->}[ur] & Y, } \]
which is possible by virtue of our assumption that $f$ is weakly left orthogonal to $g$. $\square$