Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Definition 1.4.4.4. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts and let $T$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We will say that $T$ is closed under pushouts if, for every pushout diagram

\[ \xymatrix@C =40pt@R=40pt{ A \ar [d]^{f} \ar [r] & A' \ar [d]^{f'} \\ B \ar [r] & B' } \]

in the category $\operatorname{\mathcal{C}}$ where the morphism $f$ belongs to $T$, the morphism $f'$ also belongs to $T$.