Definition 1.5.4.4. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts and let $S$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We will say that $S$ 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 $S$, the morphism $f'$ also belongs to $S$.