Definition 9.2.3.1. Let $\operatorname{\mathcal{C}}$ be a category and let $w: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We say that an object $C \in \operatorname{\mathcal{C}}$ is weakly $w$-local if, for every morphism $f: X \rightarrow C$ of $\operatorname{\mathcal{C}}$, there exists a morphism $g: Y \rightarrow C$ satisfying $g \circ w = f$. If $W$ is a collection of morphisms of $\operatorname{\mathcal{C}}$, we say that $C$ is weakly $W$-local if it is weakly $w$-local for each $w \in W$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$